[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.04,0:00:04.73,Default,,0000,0000,0000,,In this video, we'll be looking\Nat exponential functions and
Dialogue: 0,0:00:04.73,0:00:08.42,Default,,0000,0000,0000,,logarithm functions, and I'd\Nlike to start off by thinking
Dialogue: 0,0:00:08.42,0:00:13.96,Default,,0000,0000,0000,,about functions of the form F of\NX equals A to the power of X,
Dialogue: 0,0:00:13.96,0:00:17.64,Default,,0000,0000,0000,,where a is representing real\Npositive numbers. I'm going to
Dialogue: 0,0:00:17.64,0:00:22.44,Default,,0000,0000,0000,,split this up into three cases.\NFirst of all, the case when a
Dialogue: 0,0:00:22.44,0:00:27.61,Default,,0000,0000,0000,,equals 1 hour, then going to\Nlook at the case when A is more
Dialogue: 0,0:00:27.61,0:00:31.30,Default,,0000,0000,0000,,than one, and finally I'll look\Nat the case where.
Dialogue: 0,0:00:31.31,0:00:36.54,Default,,0000,0000,0000,,Is between zero and one. So\Nfirst of all.
Dialogue: 0,0:00:36.79,0:00:42.88,Default,,0000,0000,0000,,When a equals 1, this will give\Nus the function F of X equals 1
Dialogue: 0,0:00:42.88,0:00:48.97,Default,,0000,0000,0000,,to the power of X and we can see\Nthat once the power of anything
Dialogue: 0,0:00:48.97,0:00:54.65,Default,,0000,0000,0000,,is actually one. So this is the\Nlinear function F of X equals 1.
Dialogue: 0,0:00:55.30,0:01:00.30,Default,,0000,0000,0000,,So that's quite straightforward,\Nand Secondly, I'd like to look
Dialogue: 0,0:01:00.30,0:01:04.30,Default,,0000,0000,0000,,at the case where a is more than
Dialogue: 0,0:01:04.30,0:01:09.04,Default,,0000,0000,0000,,one. And to demonstrate what\Nhappens in this case, I'd like
Dialogue: 0,0:01:09.04,0:01:13.24,Default,,0000,0000,0000,,to consider a specific example.\NIn this case, I'll choose A to
Dialogue: 0,0:01:13.24,0:01:18.49,Default,,0000,0000,0000,,be equal to two, which gives us\Nthe function F of X equals 2 to
Dialogue: 0,0:01:18.49,0:01:19.89,Default,,0000,0000,0000,,the power of X.
Dialogue: 0,0:01:20.62,0:01:24.13,Default,,0000,0000,0000,,Now a good place to start with\Nthese kind of functions is to
Dialogue: 0,0:01:24.13,0:01:26.56,Default,,0000,0000,0000,,look at for some different\Nvalues of the argument.
Dialogue: 0,0:01:27.32,0:01:32.44,Default,,0000,0000,0000,,So starts off by looking at F of\N0, which is actually equal to 2
Dialogue: 0,0:01:32.44,0:01:33.80,Default,,0000,0000,0000,,to the power 0.
Dialogue: 0,0:01:34.33,0:01:38.54,Default,,0000,0000,0000,,And we know that anything to the\Npower 0 equals 1.
Dialogue: 0,0:01:38.66,0:01:41.93,Default,,0000,0000,0000,,Next, we'll look at F of one.
Dialogue: 0,0:01:41.93,0:01:44.18,Default,,0000,0000,0000,,Which is 2 to the power of 1.
Dialogue: 0,0:01:44.98,0:01:47.46,Default,,0000,0000,0000,,And two to the power of one is
Dialogue: 0,0:01:47.46,0:01:51.46,Default,,0000,0000,0000,,2. And we can look at F of
Dialogue: 0,0:01:51.46,0:01:58.87,Default,,0000,0000,0000,,two. Which is 2 squared,\Nwhich is 4. So quite
Dialogue: 0,0:01:58.87,0:02:01.67,Default,,0000,0000,0000,,straightforward, and finally F
Dialogue: 0,0:02:01.67,0:02:07.94,Default,,0000,0000,0000,,of three. Which is 2 to the\Npower of three which actually
Dialogue: 0,0:02:07.94,0:02:09.40,Default,,0000,0000,0000,,gives us 8.
Dialogue: 0,0:02:09.41,0:02:13.78,Default,,0000,0000,0000,,Also want to consider some\Nnegative arguments as well, so
Dialogue: 0,0:02:13.78,0:02:17.28,Default,,0000,0000,0000,,if we look at F of minus one.
Dialogue: 0,0:02:17.29,0:02:21.53,Default,,0000,0000,0000,,This is 2 to the power of\Nminus one. And remember when
Dialogue: 0,0:02:21.53,0:02:25.41,Default,,0000,0000,0000,,we have a negative power,\Nthat means that we have to
Dialogue: 0,0:02:25.41,0:02:28.94,Default,,0000,0000,0000,,invert our number so we\Nactually end up with one
Dialogue: 0,0:02:28.94,0:02:29.29,Default,,0000,0000,0000,,half.
Dialogue: 0,0:02:30.43,0:02:37.65,Default,,0000,0000,0000,,If you look at F of minus two\NI guess is 2 to the power of
Dialogue: 0,0:02:37.65,0:02:42.16,Default,,0000,0000,0000,,minus two. Once again, this\Nnegative power makes we've got
Dialogue: 0,0:02:42.16,0:02:48.02,Default,,0000,0000,0000,,one over 2 squared and it's 2\Nsquared is 4's actually gives us
Dialogue: 0,0:02:48.02,0:02:54.06,Default,,0000,0000,0000,,1/4. And final arguments are\Nconsider is F of minus three.
Dialogue: 0,0:02:54.79,0:03:00.39,Default,,0000,0000,0000,,Which is 2 to the power of\Nminus three, which gives us
Dialogue: 0,0:03:00.39,0:03:06.46,Default,,0000,0000,0000,,one over 2 cubed and two\Ncubes 8. So we get one 8th.
Dialogue: 0,0:03:07.51,0:03:11.13,Default,,0000,0000,0000,,I'm going to take these\Nresults now and put them into
Dialogue: 0,0:03:11.13,0:03:15.74,Default,,0000,0000,0000,,a table and we can use that\Ntable to help us plot a graph
Dialogue: 0,0:03:15.74,0:03:16.72,Default,,0000,0000,0000,,of the function.
Dialogue: 0,0:03:18.76,0:03:23.59,Default,,0000,0000,0000,,So our table, the\Nvalues of X&F of X.
Dialogue: 0,0:03:25.43,0:03:33.03,Default,,0000,0000,0000,,We just come from minus 3\N- 2 - 1 zero, 1
Dialogue: 0,0:03:33.03,0:03:36.82,Default,,0000,0000,0000,,two and three and the value
Dialogue: 0,0:03:36.82,0:03:38.09,Default,,0000,0000,0000,,hardware 1/8.
Dialogue: 0,0:03:38.59,0:03:45.06,Default,,0000,0000,0000,,1/4\N1/2 one, 2,
Dialogue: 0,0:03:45.06,0:03:48.66,Default,,0000,0000,0000,,four and eight.
Dialogue: 0,0:03:49.19,0:03:53.56,Default,,0000,0000,0000,,So we're going to plot these now\Nso we can get a graph of the
Dialogue: 0,0:03:53.56,0:03:57.53,Default,,0000,0000,0000,,function. So do RF of X
Dialogue: 0,0:03:57.53,0:04:03.78,Default,,0000,0000,0000,,axis here. And X axis\Nhorizontally. So on the X axis
Dialogue: 0,0:04:03.78,0:04:07.19,Default,,0000,0000,0000,,we need to go from minus three
Dialogue: 0,0:04:07.19,0:04:14.93,Default,,0000,0000,0000,,to +3. So minus 1\N- 2 - 3 and
Dialogue: 0,0:04:14.93,0:04:18.49,Default,,0000,0000,0000,,one. Two and three this
Dialogue: 0,0:04:18.49,0:04:23.28,Default,,0000,0000,0000,,way. And on the vertical scale,\Nthe F of X axis, we need to go
Dialogue: 0,0:04:23.28,0:04:28.97,Default,,0000,0000,0000,,up to 8. So\N12345678. Make
Dialogue: 0,0:04:28.97,0:04:31.68,Default,,0000,0000,0000,,sure we
Dialogue: 0,0:04:31.68,0:04:38.86,Default,,0000,0000,0000,,label that.\NSo now let's plot the
Dialogue: 0,0:04:38.86,0:04:39.65,Default,,0000,0000,0000,,points.
Dialogue: 0,0:04:41.04,0:04:43.01,Default,,0000,0000,0000,,Minus 3 1/8.
Dialogue: 0,0:04:43.86,0:04:46.79,Default,,0000,0000,0000,,Minus 2
Dialogue: 0,0:04:46.79,0:04:54.16,Default,,0000,0000,0000,,1/4. Minus\N1 1/2.
Dialogue: 0,0:04:55.17,0:04:56.70,Default,,0000,0000,0000,,Zero and one.
Dialogue: 0,0:04:57.63,0:05:05.04,Default,,0000,0000,0000,,12\N214
Dialogue: 0,0:05:06.66,0:05:10.30,Default,,0000,0000,0000,,And finally, three and eight.
Dialogue: 0,0:05:11.95,0:05:17.44,Default,,0000,0000,0000,,And so we need to try and draw a\Nsmooth curve through the points.
Dialogue: 0,0:05:22.24,0:05:29.04,Default,,0000,0000,0000,,And this is the graph\Nof the function F of
Dialogue: 0,0:05:29.04,0:05:34.48,Default,,0000,0000,0000,,X equals 2 to the\Npower of X.
Dialogue: 0,0:05:35.60,0:05:39.48,Default,,0000,0000,0000,,Now this is actually quite\Nclearly shows the general shape
Dialogue: 0,0:05:39.48,0:05:45.69,Default,,0000,0000,0000,,of graphs of the functions where\NF of X equals A to the X, and a
Dialogue: 0,0:05:45.69,0:05:50.34,Default,,0000,0000,0000,,is more than one. However, what\Nhappens when we vary the value
Dialogue: 0,0:05:50.34,0:05:56.02,Default,,0000,0000,0000,,of A? Well, by looking at a few\Nsketches of a few different
Dialogue: 0,0:05:56.02,0:05:57.56,Default,,0000,0000,0000,,graphs, they should become
Dialogue: 0,0:05:57.56,0:06:01.11,Default,,0000,0000,0000,,clear. So I have my axes again.
Dialogue: 0,0:06:01.81,0:06:05.53,Default,,0000,0000,0000,,F of X.
Dialogue: 0,0:06:06.31,0:06:13.27,Default,,0000,0000,0000,,X horizontally.\NWe've just spotted the graph of
Dialogue: 0,0:06:13.27,0:06:19.70,Default,,0000,0000,0000,,this function. Which one through\N1 F of X axis and this was F of
Dialogue: 0,0:06:19.70,0:06:22.52,Default,,0000,0000,0000,,X equals 2 to the power of X.
Dialogue: 0,0:06:23.04,0:06:26.71,Default,,0000,0000,0000,,If we were to look at this.
Dialogue: 0,0:06:27.47,0:06:28.22,Default,,0000,0000,0000,,Graph.
Dialogue: 0,0:06:29.60,0:06:36.96,Default,,0000,0000,0000,,This might represent F of X\Nequals 5 to the power of X.
Dialogue: 0,0:06:38.75,0:06:40.56,Default,,0000,0000,0000,,If I was look at this graph.
Dialogue: 0,0:06:43.33,0:06:49.05,Default,,0000,0000,0000,,This might represent the graph\Nof the function F of X equals 10
Dialogue: 0,0:06:49.05,0:06:51.25,Default,,0000,0000,0000,,to the power of X.
Dialogue: 0,0:06:52.12,0:06:55.60,Default,,0000,0000,0000,,So what's actually happening\Nhere? Well, for bigger values of
Dialogue: 0,0:06:55.60,0:07:01.88,Default,,0000,0000,0000,,a. We can see that the output\Nincreases more quickly as the
Dialogue: 0,0:07:01.88,0:07:02.84,Default,,0000,0000,0000,,arguments increases.
Dialogue: 0,0:07:04.15,0:07:08.11,Default,,0000,0000,0000,,Another couple of important\Npoints to notice here are first
Dialogue: 0,0:07:08.11,0:07:12.47,Default,,0000,0000,0000,,of all that every single graph\Nthat I've sketched here comes
Dialogue: 0,0:07:12.47,0:07:17.22,Default,,0000,0000,0000,,through this .0 one, and in\Nfact, regardless of our value of
Dialogue: 0,0:07:17.22,0:07:20.61,Default,,0000,0000,0000,,A. F of 0.
Dialogue: 0,0:07:21.37,0:07:24.92,Default,,0000,0000,0000,,Will equal 1 for every single\Nvalue of a.
Dialogue: 0,0:07:25.68,0:07:32.89,Default,,0000,0000,0000,,Secondly, we notice that F of\NX is always more than 0.
Dialogue: 0,0:07:33.79,0:07:37.83,Default,,0000,0000,0000,,In other words, are output for\Nthis function is always
Dialogue: 0,0:07:37.83,0:07:41.09,Default,,0000,0000,0000,,positive. As a very\Nimportant feature of these
Dialogue: 0,0:07:41.09,0:07:42.02,Default,,0000,0000,0000,,kind of functions.
Dialogue: 0,0:07:43.87,0:07:46.60,Default,,0000,0000,0000,,The last case I would like to
Dialogue: 0,0:07:46.60,0:07:53.14,Default,,0000,0000,0000,,consider. Is the case where a is\Nbetween zero and one case where
Dialogue: 0,0:07:53.14,0:07:55.89,Default,,0000,0000,0000,,a is between zero and one.
Dialogue: 0,0:07:56.49,0:08:01.49,Default,,0000,0000,0000,,To demonstrate this case, I\Nwould like it to look at a
Dialogue: 0,0:08:01.49,0:08:06.50,Default,,0000,0000,0000,,specific example. In this case I\Nwill choose a equals 1/2, so
Dialogue: 0,0:08:06.50,0:08:10.67,Default,,0000,0000,0000,,this means I'm looking at the\Nfunction F of X.
Dialogue: 0,0:08:11.21,0:08:15.90,Default,,0000,0000,0000,,Equals 1/2 to\Nthe power of X.
Dialogue: 0,0:08:16.99,0:08:20.84,Default,,0000,0000,0000,,Now with the last example, a\Ngood place to start is by
Dialogue: 0,0:08:20.84,0:08:22.45,Default,,0000,0000,0000,,looking at some different values
Dialogue: 0,0:08:22.45,0:08:27.45,Default,,0000,0000,0000,,for the arguments. So let's\Nfirst of all consider F of 0.
Dialogue: 0,0:08:28.69,0:08:33.20,Default,,0000,0000,0000,,This will give us 1/2 to the\Npower of 0.
Dialogue: 0,0:08:33.84,0:08:37.17,Default,,0000,0000,0000,,And as we said before, anything\Nto the power of 0 is one.
Dialogue: 0,0:08:37.99,0:08:41.85,Default,,0000,0000,0000,,Secondly, we look at F of one.
Dialogue: 0,0:08:42.58,0:08:46.26,Default,,0000,0000,0000,,Which is 1/2 to the power of 1.
Dialogue: 0,0:08:47.04,0:08:48.61,Default,,0000,0000,0000,,Which is equal to 1/2.
Dialogue: 0,0:08:51.02,0:08:53.32,Default,,0000,0000,0000,,Half of 2.
Dialogue: 0,0:08:53.83,0:08:57.56,Default,,0000,0000,0000,,Equals 1/2 squared.
Dialogue: 0,0:08:58.34,0:09:01.77,Default,,0000,0000,0000,,So on the top that just gives\NUS1 squared, which is one.
Dialogue: 0,0:09:02.45,0:09:08.27,Default,,0000,0000,0000,,On the bottom 2 squared, which\Nis 4. So we end up with one
Dialogue: 0,0:09:08.27,0:09:11.50,Default,,0000,0000,0000,,quarter. An F of three.
Dialogue: 0,0:09:12.33,0:09:15.89,Default,,0000,0000,0000,,Is equal to 1/2 cubed.
Dialogue: 0,0:09:17.15,0:09:22.12,Default,,0000,0000,0000,,So on the top we get one cubed\Nwhich just gives US1 and under
Dialogue: 0,0:09:22.12,0:09:27.09,Default,,0000,0000,0000,,bottom 2 cubed which gives us 8.\NSo we end up with one 8th.
Dialogue: 0,0:09:27.67,0:09:31.33,Default,,0000,0000,0000,,And as before, we also need\Nto consider some negative
Dialogue: 0,0:09:31.33,0:09:31.70,Default,,0000,0000,0000,,arguments.
Dialogue: 0,0:09:32.75,0:09:35.58,Default,,0000,0000,0000,,So half of minus one.
Dialogue: 0,0:09:36.13,0:09:39.89,Default,,0000,0000,0000,,Gives us 1/2 to the power of
Dialogue: 0,0:09:39.89,0:09:44.79,Default,,0000,0000,0000,,minus one. And remember the\Nminus sign on the power actually
Dialogue: 0,0:09:44.79,0:09:49.44,Default,,0000,0000,0000,,inverts are fraction, so we end\Nup with two over 1 to the power
Dialogue: 0,0:09:49.44,0:09:51.43,Default,,0000,0000,0000,,of 1, which is just two.
Dialogue: 0,0:09:51.97,0:09:55.08,Default,,0000,0000,0000,,F of
Dialogue: 0,0:09:55.08,0:10:02.42,Default,,0000,0000,0000,,minus 2. Gives\Nus 1/2 to the power of minus
Dialogue: 0,0:10:02.42,0:10:09.51,Default,,0000,0000,0000,,2. Which is 2 over 1 squared.\NYou can see on the top we get 2
Dialogue: 0,0:10:09.51,0:10:14.45,Default,,0000,0000,0000,,squared which is 4 and on the\Nbottom we just get one. So
Dialogue: 0,0:10:14.45,0:10:16.35,Default,,0000,0000,0000,,that's actually equal to 4.
Dialogue: 0,0:10:17.63,0:10:20.91,Default,,0000,0000,0000,,And finally, F of minus three.
Dialogue: 0,0:10:21.65,0:10:25.35,Default,,0000,0000,0000,,Is one half to the power of
Dialogue: 0,0:10:25.35,0:10:31.60,Default,,0000,0000,0000,,-3? Which means we get two over\None and we deal with the minus
Dialogue: 0,0:10:31.60,0:10:36.28,Default,,0000,0000,0000,,sign cubed. So 2 cubed in the\Ntop which is 8 and again just
Dialogue: 0,0:10:36.28,0:10:39.30,Default,,0000,0000,0000,,the one on the bottom. So that\Njust gives us 8.
Dialogue: 0,0:10:40.66,0:10:43.66,Default,,0000,0000,0000,,So once again, we're going to\Ntake these results and put them
Dialogue: 0,0:10:43.66,0:10:47.16,Default,,0000,0000,0000,,into a table so we can plot a\Ngraph. A graph of the function.
Dialogue: 0,0:10:48.54,0:10:55.43,Default,,0000,0000,0000,,So X&F of X\Nagain for our table.
Dialogue: 0,0:10:56.11,0:11:00.82,Default,,0000,0000,0000,,And we have values of the\Nargument ranging from minus
Dialogue: 0,0:11:00.82,0:11:04.59,Default,,0000,0000,0000,,three, all the way up to\Nthree again.
Dialogue: 0,0:11:06.85,0:11:12.84,Default,,0000,0000,0000,,And this time the\Nvalues where 8421.
Dialogue: 0,0:11:13.51,0:11:17.07,Default,,0000,0000,0000,,1/2 one quarter.
Dialogue: 0,0:11:17.69,0:11:19.26,Default,,0000,0000,0000,,And one 8th.
Dialogue: 0,0:11:20.26,0:11:23.08,Default,,0000,0000,0000,,So let's plot this now\Non a graph.
Dialogue: 0,0:11:24.87,0:11:31.46,Default,,0000,0000,0000,,So vertically we get F of X and\Na horizontal axis. We've got X
Dialogue: 0,0:11:31.46,0:11:35.70,Default,,0000,0000,0000,,and we're going from minus three\Nto three again.
Dialogue: 0,0:11:37.83,0:11:41.27,Default,,0000,0000,0000,,So minus 1 - 2 -
Dialogue: 0,0:11:41.27,0:11:44.20,Default,,0000,0000,0000,,3. 123
Dialogue: 0,0:11:44.20,0:11:51.36,Default,,0000,0000,0000,,here. And\Nthen we're going up
Dialogue: 0,0:11:51.36,0:11:57.63,Default,,0000,0000,0000,,to 8 on the\Nvertical axis, 12345678.
Dialogue: 0,0:11:57.94,0:12:03.51,Default,,0000,0000,0000,,So let's\Nplot the
Dialogue: 0,0:12:03.51,0:12:08.28,Default,,0000,0000,0000,,points. First point\Nis minus three 8.
Dialogue: 0,0:12:10.28,0:12:13.25,Default,,0000,0000,0000,,Which is around about here.
Dialogue: 0,0:12:14.02,0:12:16.16,Default,,0000,0000,0000,,Minus two and four.
Dialogue: 0,0:12:18.82,0:12:22.01,Default,,0000,0000,0000,,But here. Minus one and two.
Dialogue: 0,0:12:23.53,0:12:26.44,Default,,0000,0000,0000,,But the. Zero and one.
Dialogue: 0,0:12:27.14,0:12:35.23,Default,,0000,0000,0000,,One 1/2.\N2 one quarter.
Dialogue: 0,0:12:35.99,0:12:39.52,Default,,0000,0000,0000,,And finally 3 1/8.
Dialogue: 0,0:12:40.07,0:12:44.89,Default,,0000,0000,0000,,As with our previous example, we\Nneed to try and draw a smooth
Dialogue: 0,0:12:44.89,0:12:46.38,Default,,0000,0000,0000,,curve through the points.
Dialogue: 0,0:12:47.61,0:12:48.53,Default,,0000,0000,0000,,So.
Dialogue: 0,0:12:56.87,0:13:03.91,Default,,0000,0000,0000,,And this represents the function\NF of X equals 1/2 to
Dialogue: 0,0:13:03.91,0:13:06.47,Default,,0000,0000,0000,,the power of X.
Dialogue: 0,0:13:07.08,0:13:10.65,Default,,0000,0000,0000,,And actually this demonstrates\Nthe general shape for functions
Dialogue: 0,0:13:10.65,0:13:17.40,Default,,0000,0000,0000,,of the form F of X equals A to\Nthe X when A is between zero and
Dialogue: 0,0:13:17.40,0:13:21.77,Default,,0000,0000,0000,,one. But what happens when we\Nvary a within those boundaries?
Dialogue: 0,0:13:22.38,0:13:25.15,Default,,0000,0000,0000,,Well, sketching a few\Ngraphs of this function
Dialogue: 0,0:13:25.15,0:13:26.88,Default,,0000,0000,0000,,will help us to see.
Dialogue: 0,0:13:28.20,0:13:31.09,Default,,0000,0000,0000,,Just do some axes again.
Dialogue: 0,0:13:31.71,0:13:33.37,Default,,0000,0000,0000,,So we've got F of X.
Dialogue: 0,0:13:36.12,0:13:39.31,Default,,0000,0000,0000,,And X. We've just seen.
Dialogue: 0,0:13:44.03,0:13:46.11,Default,,0000,0000,0000,,This curve, which was.
Dialogue: 0,0:13:46.97,0:13:52.72,Default,,0000,0000,0000,,F of X equals 1/2 to\Nthe power of X.
Dialogue: 0,0:13:53.60,0:13:55.93,Default,,0000,0000,0000,,We might have seen.
Dialogue: 0,0:14:02.12,0:14:04.35,Default,,0000,0000,0000,,This would have represented.
Dialogue: 0,0:14:06.22,0:14:08.56,Default,,0000,0000,0000,,F of X equals.
Dialogue: 0,0:14:09.40,0:14:14.28,Default,,0000,0000,0000,,1/5 to the power of X or we may\Neven have seen.
Dialogue: 0,0:14:15.16,0:14:16.46,Default,,0000,0000,0000,,Something like this?
Dialogue: 0,0:14:21.30,0:14:27.20,Default,,0000,0000,0000,,Which would have represented\Nmaybe F of X equals 110th to the
Dialogue: 0,0:14:27.20,0:14:28.68,Default,,0000,0000,0000,,power of X.
Dialogue: 0,0:14:30.02,0:14:36.71,Default,,0000,0000,0000,,So we can see that for bigger\Nvalues of a this is. We come
Dialogue: 0,0:14:36.71,0:14:41.49,Default,,0000,0000,0000,,down here. The output decreases\Nmore slowly as the arguments
Dialogue: 0,0:14:41.49,0:14:46.56,Default,,0000,0000,0000,,increases. And a few important\Npoints to notice here. First of
Dialogue: 0,0:14:46.56,0:14:50.62,Default,,0000,0000,0000,,all, as with the previous\Nexample, F of 0 equals 1
Dialogue: 0,0:14:50.62,0:14:55.78,Default,,0000,0000,0000,,regardless of the value of A and\Nin fact, as long as a is
Dialogue: 0,0:14:55.78,0:14:59.11,Default,,0000,0000,0000,,positive and real, this will\Nalways be the case.
Dialogue: 0,0:15:00.90,0:15:05.67,Default,,0000,0000,0000,,Second thing to notice, as with\Nthe previous example, is that
Dialogue: 0,0:15:05.67,0:15:10.88,Default,,0000,0000,0000,,our output F of X is always\Npositive, so output is always
Dialogue: 0,0:15:10.88,0:15:12.18,Default,,0000,0000,0000,,more than 0.
Dialogue: 0,0:15:15.02,0:15:19.38,Default,,0000,0000,0000,,So now we've looked at what\Nhappens for all the different
Dialogue: 0,0:15:19.38,0:15:22.15,Default,,0000,0000,0000,,values of a when A is positive
Dialogue: 0,0:15:22.15,0:15:24.74,Default,,0000,0000,0000,,and real. No one\Ninteresting thing you
Dialogue: 0,0:15:24.74,0:15:27.60,Default,,0000,0000,0000,,might have noticed is\Nthis. We've got some
Dialogue: 0,0:15:27.60,0:15:31.18,Default,,0000,0000,0000,,symmetry going on here. If\NI actually just put some
Dialogue: 0,0:15:31.18,0:15:32.62,Default,,0000,0000,0000,,axes down again here.
Dialogue: 0,0:15:33.79,0:15:37.76,Default,,0000,0000,0000,,So F of X find X you'll\Nremember.
Dialogue: 0,0:15:39.60,0:15:40.97,Default,,0000,0000,0000,,This curve here.
Dialogue: 0,0:15:42.51,0:15:47.84,Default,,0000,0000,0000,,Passing through the point one\Nwas F of X equals 2 to the power
Dialogue: 0,0:15:47.84,0:15:52.66,Default,,0000,0000,0000,,of X. And also\Nthis curve here.
Dialogue: 0,0:15:56.17,0:16:02.91,Default,,0000,0000,0000,,Was F of X equals 1/2\Nto the power of X?
Dialogue: 0,0:16:03.90,0:16:08.95,Default,,0000,0000,0000,,And we can see this link it\N'cause one of these graphs is a
Dialogue: 0,0:16:08.95,0:16:14.01,Default,,0000,0000,0000,,reflection of the other in the F\Nof X axis. And in fact this
Dialogue: 0,0:16:14.01,0:16:19.78,Default,,0000,0000,0000,,could have been 1/5 to the X and\Nthis could have been 5 to the X
Dialogue: 0,0:16:19.78,0:16:20.87,Default,,0000,0000,0000,,'cause generally speaking.
Dialogue: 0,0:16:20.87,0:16:23.63,Default,,0000,0000,0000,,F of X equal to.
Dialogue: 0,0:16:24.23,0:16:25.98,Default,,0000,0000,0000,,AX.
Dialogue: 0,0:16:27.04,0:16:29.73,Default,,0000,0000,0000,,Is a
Dialogue: 0,0:16:29.73,0:16:36.100,Default,,0000,0000,0000,,reflection. Of F of\NX equals 1 over
Dialogue: 0,0:16:36.100,0:16:43.76,Default,,0000,0000,0000,,8 to the X\Nand that is in
Dialogue: 0,0:16:43.76,0:16:47.14,Default,,0000,0000,0000,,the F of X
Dialogue: 0,0:16:47.14,0:16:54.79,Default,,0000,0000,0000,,axis. And one over 8 to the\NX can also be written as A to
Dialogue: 0,0:16:54.79,0:16:56.19,Default,,0000,0000,0000,,the minus X.
Dialogue: 0,0:16:57.38,0:16:59.90,Default,,0000,0000,0000,,So that's an interesting point
Dialogue: 0,0:16:59.90,0:17:06.18,Default,,0000,0000,0000,,to note. Now you might recall\Nfrom Chapter 2.3 that the
Dialogue: 0,0:17:06.18,0:17:07.62,Default,,0000,0000,0000,,exponential number E.
Dialogue: 0,0:17:08.31,0:17:11.35,Default,,0000,0000,0000,,Is approximately
Dialogue: 0,0:17:11.35,0:17:16.48,Default,,0000,0000,0000,,2.718. Which means it falls\Ninto the first category, where
Dialogue: 0,0:17:16.48,0:17:20.62,Default,,0000,0000,0000,,a is more than one. If we're\Ngoing to consider the function
Dialogue: 0,0:17:20.62,0:17:25.11,Default,,0000,0000,0000,,F of X equals E to the X, which\Nis the exponential function.
Dialogue: 0,0:17:27.65,0:17:30.34,Default,,0000,0000,0000,,So if I do some axes again.
Dialogue: 0,0:17:30.91,0:17:32.80,Default,,0000,0000,0000,,F of X vertically.
Dialogue: 0,0:17:34.06,0:17:35.97,Default,,0000,0000,0000,,An ex horizontally.
Dialogue: 0,0:17:39.34,0:17:40.58,Default,,0000,0000,0000,,This graph here.
Dialogue: 0,0:17:41.73,0:17:47.20,Default,,0000,0000,0000,,Might represent F of X equals E\Nto the X and this is what the
Dialogue: 0,0:17:47.20,0:17:48.30,Default,,0000,0000,0000,,exponential function actually
Dialogue: 0,0:17:48.30,0:17:53.46,Default,,0000,0000,0000,,looks like. I might also like to\Nconsider the function F of X
Dialogue: 0,0:17:53.46,0:17:55.53,Default,,0000,0000,0000,,equals E to the minus X.
Dialogue: 0,0:17:56.19,0:17:57.58,Default,,0000,0000,0000,,As we've just discovered.
Dialogue: 0,0:17:59.62,0:18:06.10,Default,,0000,0000,0000,,That is a reflection in the F of\NX axis, so this will be F of X
Dialogue: 0,0:18:06.10,0:18:10.67,Default,,0000,0000,0000,,equals E to the minus X. And\Nremember this important .1 on
Dialogue: 0,0:18:10.67,0:18:12.57,Default,,0000,0000,0000,,the F of X axis.
Dialogue: 0,0:18:13.78,0:18:17.22,Default,,0000,0000,0000,,So look that exponential\Nfunctions and we've looked the
Dialogue: 0,0:18:17.22,0:18:19.89,Default,,0000,0000,0000,,functions of the form A to the
Dialogue: 0,0:18:19.89,0:18:22.88,Default,,0000,0000,0000,,X. At my work to consider
Dialogue: 0,0:18:22.88,0:18:26.84,Default,,0000,0000,0000,,logarithm functions. And\Nlogarithm functions take the
Dialogue: 0,0:18:26.84,0:18:32.93,Default,,0000,0000,0000,,form F of X equals the log\Nof X. So particular base. In
Dialogue: 0,0:18:32.93,0:18:34.34,Default,,0000,0000,0000,,this case a.
Dialogue: 0,0:18:35.69,0:18:39.30,Default,,0000,0000,0000,,And as with the previous\Nexample, I'd like to split my
Dialogue: 0,0:18:39.30,0:18:42.91,Default,,0000,0000,0000,,analysis of this into three\Nparts. First of all, looking at
Dialogue: 0,0:18:42.91,0:18:44.22,Default,,0000,0000,0000,,when a is one.
Dialogue: 0,0:18:44.89,0:18:48.93,Default,,0000,0000,0000,,Second of all, looking at when\Nthere is more than one, and
Dialogue: 0,0:18:48.93,0:18:52.98,Default,,0000,0000,0000,,finally when a is a number\Nbetween zero and one, and as
Dialogue: 0,0:18:52.98,0:18:57.02,Default,,0000,0000,0000,,with the previous example, a is\Nonly going to be positive real
Dialogue: 0,0:18:57.02,0:19:01.99,Default,,0000,0000,0000,,numbers. So first of all, what\Nhappens when a equals 1?
Dialogue: 0,0:19:02.97,0:19:08.38,Default,,0000,0000,0000,,Well, this means we'll get a\Nfunction F of X equals the log
Dialogue: 0,0:19:08.38,0:19:10.46,Default,,0000,0000,0000,,of X to base one.
Dialogue: 0,0:19:11.04,0:19:14.01,Default,,0000,0000,0000,,Remember, this is equivalent.
Dialogue: 0,0:19:14.93,0:19:16.70,Default,,0000,0000,0000,,To say that this number one.
Dialogue: 0,0:19:17.22,0:19:20.01,Default,,0000,0000,0000,,To some power.
Dialogue: 0,0:19:20.05,0:19:26.03,Default,,0000,0000,0000,,F of X is equal to X, just\Nlike earlier on, this is
Dialogue: 0,0:19:26.03,0:19:32.66,Default,,0000,0000,0000,,equivalent. Generally. 2A\Nto the power of F of X
Dialogue: 0,0:19:32.66,0:19:33.66,Default,,0000,0000,0000,,equals X.
Dialogue: 0,0:19:35.18,0:19:40.26,Default,,0000,0000,0000,,So when we look at this, we can\Nsee that we can only get
Dialogue: 0,0:19:40.26,0:19:43.53,Default,,0000,0000,0000,,solutions when we consider the\Narguments X equals 1.
Dialogue: 0,0:19:44.13,0:19:48.67,Default,,0000,0000,0000,,And in fact, if we look at the\Narguments X equals 1, there is
Dialogue: 0,0:19:48.67,0:19:52.88,Default,,0000,0000,0000,,an infinite number of answers be\Ncause one to any power will give
Dialogue: 0,0:19:52.88,0:19:56.77,Default,,0000,0000,0000,,us one. So actually this is not\Na valid function because we've
Dialogue: 0,0:19:56.77,0:20:00.33,Default,,0000,0000,0000,,got many outputs for just one\Nsingle input. So that's what
Dialogue: 0,0:20:00.33,0:20:04.54,Default,,0000,0000,0000,,happens for a equals 1 would\Nhappen for a is more than one.
Dialogue: 0,0:20:05.24,0:20:09.33,Default,,0000,0000,0000,,Let's look at the case when I is\Nmore than one.
Dialogue: 0,0:20:09.87,0:20:15.04,Default,,0000,0000,0000,,So this means we're looking at\Nthe function F of X equals and
Dialogue: 0,0:20:15.04,0:20:19.82,Default,,0000,0000,0000,,in this case I will choose a\Nequals 2 again to demonstrate
Dialogue: 0,0:20:19.82,0:20:25.93,Default,,0000,0000,0000,,what's happening. So F of X\Nequals the log of X to base 2.
Dialogue: 0,0:20:26.61,0:20:29.35,Default,,0000,0000,0000,,And remember, this is
Dialogue: 0,0:20:29.35,0:20:33.95,Default,,0000,0000,0000,,equivalent. The same 2 to the\Npower of F of X.
Dialogue: 0,0:20:34.45,0:20:40.03,Default,,0000,0000,0000,,Equals X. So as with all\Nfunctions, when we get to this
Dialogue: 0,0:20:40.03,0:20:43.29,Default,,0000,0000,0000,,kind of situation, we want to\Nstart looking at some different
Dialogue: 0,0:20:43.29,0:20:47.13,Default,,0000,0000,0000,,values for the argument to help\Nus plot a graph of the function.
Dialogue: 0,0:20:47.82,0:20:51.36,Default,,0000,0000,0000,,However, before we do that, we\Nmight just want to take a closer
Dialogue: 0,0:20:51.36,0:20:52.99,Default,,0000,0000,0000,,look at what's going on here.
Dialogue: 0,0:20:54.29,0:20:59.99,Default,,0000,0000,0000,,Because 2 to the power of F of\NX. In other words, two to some
Dialogue: 0,0:20:59.99,0:21:04.93,Default,,0000,0000,0000,,power can never ever be negative\Nor 0. This is a positive number.
Dialogue: 0,0:21:04.93,0:21:08.73,Default,,0000,0000,0000,,Whatever we choose, and since\Nthat's more than zero, and
Dialogue: 0,0:21:08.73,0:21:13.67,Default,,0000,0000,0000,,that's equal to X, this means\Nthat X must be more than 0.
Dialogue: 0,0:21:14.23,0:21:16.81,Default,,0000,0000,0000,,And X represents our\Narguments, which means I'm
Dialogue: 0,0:21:16.81,0:21:19.72,Default,,0000,0000,0000,,only going to look at\Npositive arguments for this
Dialogue: 0,0:21:19.72,0:21:20.04,Default,,0000,0000,0000,,reason.
Dialogue: 0,0:21:21.46,0:21:25.27,Default,,0000,0000,0000,,So start off by looking\Nat F of one.
Dialogue: 0,0:21:26.47,0:21:30.06,Default,,0000,0000,0000,,Now F of one gives us.
Dialogue: 0,0:21:30.14,0:21:33.32,Default,,0000,0000,0000,,The log of one to base 2.
Dialogue: 0,0:21:34.48,0:21:37.89,Default,,0000,0000,0000,,North actually means remember.
Dialogue: 0,0:21:38.44,0:21:44.82,Default,,0000,0000,0000,,Is that two to some power? Half\Nof one is equal to 1, so two to
Dialogue: 0,0:21:44.82,0:21:50.41,Default,,0000,0000,0000,,what power will give me one?\NWell, it must be 2 to the power
Dialogue: 0,0:21:50.41,0:21:54.80,Default,,0000,0000,0000,,zero. Give me one because\Nanything to the power zero gives
Dialogue: 0,0:21:54.80,0:22:00.78,Default,,0000,0000,0000,,me one. So this half of one must\Nbe 0. So therefore F of one
Dialogue: 0,0:22:00.78,0:22:07.20,Default,,0000,0000,0000,,equals 0. Next I look\Nat F of two.
Dialogue: 0,0:22:07.99,0:22:10.75,Default,,0000,0000,0000,,And this is means that we've got\Nthe log of two.
Dialogue: 0,0:22:11.54,0:22:12.62,Default,,0000,0000,0000,,To base 2.
Dialogue: 0,0:22:13.22,0:22:17.06,Default,,0000,0000,0000,,And remember that this is\Nequivalent to saying that two to
Dialogue: 0,0:22:17.06,0:22:22.29,Default,,0000,0000,0000,,some power. In this case F of\Ntwo is equal to two SO2 to what
Dialogue: 0,0:22:22.29,0:22:24.04,Default,,0000,0000,0000,,power will give Me 2?
Dialogue: 0,0:22:24.54,0:22:29.84,Default,,0000,0000,0000,,Well, we know that 2 to the\Npower of one will give Me 2, so
Dialogue: 0,0:22:29.84,0:22:32.66,Default,,0000,0000,0000,,this F of two must be equal to
Dialogue: 0,0:22:32.66,0:22:39.82,Default,,0000,0000,0000,,1. And also going to look\Nat F of F of four.
Dialogue: 0,0:22:40.43,0:22:47.31,Default,,0000,0000,0000,,Now F of four means that\Nwe've got the log of four
Dialogue: 0,0:22:47.31,0:22:50.74,Default,,0000,0000,0000,,to base two. Remember, this is
Dialogue: 0,0:22:50.74,0:22:53.03,Default,,0000,0000,0000,,equivalent. Just saying that
Dialogue: 0,0:22:53.03,0:22:57.59,Default,,0000,0000,0000,,we've got. Two to some\Npower. In this case, F
Dialogue: 0,0:22:57.59,0:23:00.01,Default,,0000,0000,0000,,of four is equal to 4.
Dialogue: 0,0:23:01.12,0:23:03.88,Default,,0000,0000,0000,,Now 2 to what power will give me
Dialogue: 0,0:23:03.88,0:23:09.53,Default,,0000,0000,0000,,4? Well, it's actually going to\Nbe 2 squared, so it must be 2 to
Dialogue: 0,0:23:09.53,0:23:13.98,Default,,0000,0000,0000,,the power of two. Will give me\N4, so therefore we can see that
Dialogue: 0,0:23:13.98,0:23:16.21,Default,,0000,0000,0000,,F of four will actually give Me
Dialogue: 0,0:23:16.21,0:23:21.84,Default,,0000,0000,0000,,2. I also want to consider\Nsome fractional arguments here
Dialogue: 0,0:23:21.84,0:23:23.91,Default,,0000,0000,0000,,between zero and one.
Dialogue: 0,0:23:24.65,0:23:27.74,Default,,0000,0000,0000,,So if I look at for example.
Dialogue: 0,0:23:28.41,0:23:30.85,Default,,0000,0000,0000,,F of 1/2.
Dialogue: 0,0:23:31.63,0:23:37.71,Default,,0000,0000,0000,,OK, this means we've got the log\Nof 1/2 to the base 2.
Dialogue: 0,0:23:38.42,0:23:41.40,Default,,0000,0000,0000,,And this is equivalent\Nto saying.
Dialogue: 0,0:23:42.58,0:23:49.63,Default,,0000,0000,0000,,The two to some power. In this\Ncase F of 1/2 equals 1/2.
Dialogue: 0,0:23:50.81,0:23:53.97,Default,,0000,0000,0000,,Now this time it's a little bit\Nmore tricky to see actually
Dialogue: 0,0:23:53.97,0:23:55.02,Default,,0000,0000,0000,,what's going on here.
Dialogue: 0,0:23:55.81,0:24:02.12,Default,,0000,0000,0000,,But remember, we can write 1/2\Nas 2 to the power of minus one.
Dialogue: 0,0:24:02.85,0:24:06.03,Default,,0000,0000,0000,,Because remember about that\Nminus power that we talked about
Dialogue: 0,0:24:06.03,0:24:10.48,Default,,0000,0000,0000,,earlier on. So 2 to the power of\Nsampling equals 2 to the power
Dialogue: 0,0:24:10.48,0:24:13.34,Default,,0000,0000,0000,,of minus one. This something\Nmust be minus one.
Dialogue: 0,0:24:13.98,0:24:20.71,Default,,0000,0000,0000,,So therefore. Half\Nof 1/2 actually equals minus
Dialogue: 0,0:24:20.71,0:24:28.16,Default,,0000,0000,0000,,one. And the final argument\NI want to consider is F of
Dialogue: 0,0:24:28.16,0:24:34.49,Default,,0000,0000,0000,,1/4. And this gives us\Nthe log of 1/4.
Dialogue: 0,0:24:34.54,0:24:35.66,Default,,0000,0000,0000,,So the base 2.
Dialogue: 0,0:24:36.16,0:24:39.93,Default,,0000,0000,0000,,So this is equivalent to saying.
Dialogue: 0,0:24:40.60,0:24:48.33,Default,,0000,0000,0000,,That two to some power. In this\Ncase F of 1/4 is equal to
Dialogue: 0,0:24:48.33,0:24:52.59,Default,,0000,0000,0000,,1/4. And once again, it's not an\Neasy step just to see exactly
Dialogue: 0,0:24:52.59,0:24:56.28,Default,,0000,0000,0000,,what's going on here. Straight\Naway. We want to try and rewrite
Dialogue: 0,0:24:56.28,0:24:57.51,Default,,0000,0000,0000,,this right hand side.
Dialogue: 0,0:24:58.04,0:25:03.74,Default,,0000,0000,0000,,Now one over 4 is the same as\None over 2 squared. +2 squared
Dialogue: 0,0:25:03.74,0:25:09.03,Default,,0000,0000,0000,,is just the same as four, and\Nremember about the minus sign so
Dialogue: 0,0:25:09.03,0:25:15.54,Default,,0000,0000,0000,,we can put that onto the top and\Nwe get 2 to the power of minus
Dialogue: 0,0:25:15.54,0:25:21.24,Default,,0000,0000,0000,,two. So here 2 to the power of\Nsomething equals 2 to the power
Dialogue: 0,0:25:21.24,0:25:25.72,Default,,0000,0000,0000,,of minus two. That something\Nmust be minus two, so therefore
Dialogue: 0,0:25:25.72,0:25:28.16,Default,,0000,0000,0000,,F of 1/4 equals minus 2.
Dialogue: 0,0:25:29.21,0:25:33.08,Default,,0000,0000,0000,,So what we're going to do now?\NWe're going to put these results
Dialogue: 0,0:25:33.08,0:25:36.66,Default,,0000,0000,0000,,into a table so we can plot a\Ngraph of the function.
Dialogue: 0,0:25:37.52,0:25:43.56,Default,,0000,0000,0000,,So we've got\NX&F of X.
Dialogue: 0,0:25:44.42,0:25:50.95,Default,,0000,0000,0000,,Now X values we\Nchose. We had one
Dialogue: 0,0:25:50.95,0:25:54.14,Default,,0000,0000,0000,,quarter. We had one half.
Dialogue: 0,0:25:54.93,0:25:58.79,Default,,0000,0000,0000,,1, two and four.
Dialogue: 0,0:25:59.30,0:26:05.34,Default,,0000,0000,0000,,And the corresponding outputs\NHere were minus 2 - 1
Dialogue: 0,0:26:05.34,0:26:07.76,Default,,0000,0000,0000,,zero one and two.
Dialogue: 0,0:26:08.49,0:26:12.28,Default,,0000,0000,0000,,So let's look at plotting the\Ngraph of this function now, so
Dialogue: 0,0:26:12.28,0:26:14.18,Default,,0000,0000,0000,,as before, we want some axes
Dialogue: 0,0:26:14.18,0:26:17.80,Default,,0000,0000,0000,,here. So F of
Dialogue: 0,0:26:17.80,0:26:21.37,Default,,0000,0000,0000,,X vertically. And X
Dialogue: 0,0:26:21.37,0:26:28.04,Default,,0000,0000,0000,,horizontally. Horizontally we\Nneed to go from
Dialogue: 0,0:26:28.04,0:26:31.60,Default,,0000,0000,0000,,one 3:45, so just
Dialogue: 0,0:26:31.60,0:26:37.61,Default,,0000,0000,0000,,go 1234. Includes everything we\Nneed and vertically we need to
Dialogue: 0,0:26:37.61,0:26:40.42,Default,,0000,0000,0000,,go from minus 2 + 2.
Dialogue: 0,0:26:40.52,0:26:42.52,Default,,0000,0000,0000,,So minus 1 - 2.
Dialogue: 0,0:26:43.16,0:26:48.33,Default,,0000,0000,0000,,I'm one and two, so let's\Nput the points.
Dialogue: 0,0:26:49.16,0:26:56.100,Default,,0000,0000,0000,,1/4 and minus 2. First of\Nall, 1/4 minus two 1/2 -
Dialogue: 0,0:26:56.100,0:27:03.90,Default,,0000,0000,0000,,1. So it's 1/2\N- 1 one and 0.
Dialogue: 0,0:27:04.86,0:27:06.04,Default,,0000,0000,0000,,21
Dialogue: 0,0:27:07.24,0:27:11.07,Default,,0000,0000,0000,,And finally, four and two.
Dialogue: 0,0:27:11.94,0:27:15.67,Default,,0000,0000,0000,,So now we want to try and\Ndraw a smooth curve through
Dialogue: 0,0:27:15.67,0:27:19.09,Default,,0000,0000,0000,,the point so we can see the\Ngraph of the function.
Dialogue: 0,0:27:20.34,0:27:27.99,Default,,0000,0000,0000,,Excellent, so this\Nis F of
Dialogue: 0,0:27:27.99,0:27:35.64,Default,,0000,0000,0000,,X equals the\Nlog of X.
Dialogue: 0,0:27:35.64,0:27:39.76,Default,,0000,0000,0000,,These two. And actually, this\Nrepresents the general shape for
Dialogue: 0,0:27:39.76,0:27:44.78,Default,,0000,0000,0000,,functions of the form F of X\Nequals the log of X to base a
Dialogue: 0,0:27:44.78,0:27:49.14,Default,,0000,0000,0000,,when A is more than one. But\Nwhat happens as we very well,
Dialogue: 0,0:27:49.14,0:27:53.50,Default,,0000,0000,0000,,let's have a look at a few\Nsketches of some graphs of some
Dialogue: 0,0:27:53.50,0:27:56.84,Default,,0000,0000,0000,,different functions and that\Nshould help us to see what's
Dialogue: 0,0:27:56.84,0:28:02.59,Default,,0000,0000,0000,,going on. So if I have\NF of X vertically.
Dialogue: 0,0:28:03.41,0:28:06.96,Default,,0000,0000,0000,,And X horizontally.
Dialogue: 0,0:28:07.92,0:28:09.54,Default,,0000,0000,0000,,We've just seen.
Dialogue: 0,0:28:11.28,0:28:14.75,Default,,0000,0000,0000,,F of
Dialogue: 0,0:28:14.75,0:28:20.95,Default,,0000,0000,0000,,X equals. Log\Nof X to base 2.
Dialogue: 0,0:28:22.47,0:28:24.15,Default,,0000,0000,0000,,And if I was to draw this.
Dialogue: 0,0:28:27.30,0:28:28.87,Default,,0000,0000,0000,,This might represent.
Dialogue: 0,0:28:29.57,0:28:36.24,Default,,0000,0000,0000,,F of X equals the log of X\Nto the base E. Remember E being
Dialogue: 0,0:28:36.24,0:28:39.80,Default,,0000,0000,0000,,the number 2.718, the\Nexponential number and actually
Dialogue: 0,0:28:39.80,0:28:44.26,Default,,0000,0000,0000,,this is called the natural log\Nand is sometimes written.
Dialogue: 0,0:28:44.90,0:28:46.01,Default,,0000,0000,0000,,LNX
Dialogue: 0,0:28:47.32,0:28:49.74,Default,,0000,0000,0000,,And finally, it might have.
Dialogue: 0,0:28:53.15,0:28:56.81,Default,,0000,0000,0000,,F of X equals.
Dialogue: 0,0:28:57.36,0:29:02.44,Default,,0000,0000,0000,,Log of AXA Base 5 maybe so we\Ncan see that what's happening
Dialogue: 0,0:29:02.44,0:29:04.79,Default,,0000,0000,0000,,here for bigger values of a.
Dialogue: 0,0:29:05.91,0:29:09.34,Default,,0000,0000,0000,,The output is increasing more
Dialogue: 0,0:29:09.34,0:29:11.94,Default,,0000,0000,0000,,slowly. As the arguments
Dialogue: 0,0:29:11.94,0:29:17.36,Default,,0000,0000,0000,,increases. Now a few important\Npoints to notice here, the first
Dialogue: 0,0:29:17.36,0:29:22.00,Default,,0000,0000,0000,,one. It's a notice this point\Nhere, this one on the X axis.
Dialogue: 0,0:29:22.86,0:29:25.55,Default,,0000,0000,0000,,Regardless of our value of A.
Dialogue: 0,0:29:26.10,0:29:32.99,Default,,0000,0000,0000,,F of one will always be 0.\NThat's true for all values of a
Dialogue: 0,0:29:32.99,0:29:36.77,Default,,0000,0000,0000,,here. Second thing to notice is\Nsomething we touched upon
Dialogue: 0,0:29:36.77,0:29:39.66,Default,,0000,0000,0000,,earlier on is just point about\Nthe arguments always being
Dialogue: 0,0:29:39.66,0:29:42.84,Default,,0000,0000,0000,,positive and we can see this\Ngraphically. Here we see we've
Dialogue: 0,0:29:42.84,0:29:46.31,Default,,0000,0000,0000,,got no points to the left of the\NF of X axis.
Dialogue: 0,0:29:47.02,0:29:50.87,Default,,0000,0000,0000,,And so X is always more than
Dialogue: 0,0:29:50.87,0:29:55.73,Default,,0000,0000,0000,,0. The final case\NI want to look at.
Dialogue: 0,0:29:56.93,0:30:02.83,Default,,0000,0000,0000,,Is the case worth a is between\Nzero and one. And to demonstrate
Dialogue: 0,0:30:02.83,0:30:08.28,Default,,0000,0000,0000,,this case I will look at the\Nspecific example where a equals
Dialogue: 0,0:30:08.28,0:30:14.05,Default,,0000,0000,0000,,1/2. And if I equals 1/2, the\Nfunction we're going to be
Dialogue: 0,0:30:14.05,0:30:20.47,Default,,0000,0000,0000,,looking at is F of X equals the\Nlog of X to the base 1/2.
Dialogue: 0,0:30:21.41,0:30:27.11,Default,,0000,0000,0000,,So remember we can rewrite this.\NThis is equivalent to saying
Dialogue: 0,0:30:27.11,0:30:34.36,Default,,0000,0000,0000,,that one half to some power. In\Nthis case F of X is equal
Dialogue: 0,0:30:34.36,0:30:38.72,Default,,0000,0000,0000,,to X. Answer the previous\Nexample. We need to think
Dialogue: 0,0:30:38.72,0:30:41.56,Default,,0000,0000,0000,,carefully about which arguments\Nwe're going to consider now.
Dialogue: 0,0:30:42.10,0:30:47.66,Default,,0000,0000,0000,,Because 1/2 to any power will\Nalways give me a positive
Dialogue: 0,0:30:47.66,0:30:52.70,Default,,0000,0000,0000,,number. In other words, 1/2. So\Nthe F of X.
Dialogue: 0,0:30:53.22,0:30:54.94,Default,,0000,0000,0000,,Is always more than 0.
Dialogue: 0,0:30:55.72,0:30:59.53,Default,,0000,0000,0000,,And since this is equal to X,\Nthis means that X is more than
Dialogue: 0,0:30:59.53,0:31:02.59,Default,,0000,0000,0000,,0. And so many going to consider
Dialogue: 0,0:31:02.59,0:31:07.98,Default,,0000,0000,0000,,positive arguments. So first of\Nall, I can set up F of one.
Dialogue: 0,0:31:08.75,0:31:15.13,Default,,0000,0000,0000,,Half of 1 means that we've\Ngot the log of one to the
Dialogue: 0,0:31:15.13,0:31:16.61,Default,,0000,0000,0000,,base of 1/2.
Dialogue: 0,0:31:17.78,0:31:19.60,Default,,0000,0000,0000,,Remember, this is equivalent.
Dialogue: 0,0:31:20.65,0:31:28.11,Default,,0000,0000,0000,,Just saying.\N1/2 to some power. In this case,
Dialogue: 0,0:31:28.11,0:31:31.61,Default,,0000,0000,0000,,F of one is equal to 1.
Dialogue: 0,0:31:33.77,0:31:38.11,Default,,0000,0000,0000,,So how does this work? 1/2 to\Nsome power equals 1. Remember
Dialogue: 0,0:31:38.11,0:31:42.46,Default,,0000,0000,0000,,anything to the power of 0\Nequals 1, so F of one.
Dialogue: 0,0:31:43.00,0:31:46.14,Default,,0000,0000,0000,,Must be 0.
Dialogue: 0,0:31:46.14,0:31:52.03,Default,,0000,0000,0000,,Secondly,\NF of two.
Dialogue: 0,0:31:52.97,0:31:55.38,Default,,0000,0000,0000,,This gives us the log of two.
Dialogue: 0,0:31:55.89,0:31:57.68,Default,,0000,0000,0000,,So the base 1/2.
Dialogue: 0,0:31:58.37,0:32:00.48,Default,,0000,0000,0000,,Remember, this is equivalent to
Dialogue: 0,0:32:00.48,0:32:06.61,Default,,0000,0000,0000,,saying. That we've got 1/2 to\Nsome power. In this case F of
Dialogue: 0,0:32:06.61,0:32:10.43,Default,,0000,0000,0000,,two. Is equal to 2.
Dialogue: 0,0:32:11.12,0:32:15.38,Default,,0000,0000,0000,,So how do we find out what this\Npowers got to be?
Dialogue: 0,0:32:15.92,0:32:19.97,Default,,0000,0000,0000,,Well, we want to look at\Nrewriting this number 2, and
Dialogue: 0,0:32:19.97,0:32:23.65,Default,,0000,0000,0000,,this is where the minus negative\Npowers come in useful.
Dialogue: 0,0:32:24.39,0:32:29.82,Default,,0000,0000,0000,,So we can actually write this as\N1/2 to the power of minus one.
Dialogue: 0,0:32:30.50,0:32:36.50,Default,,0000,0000,0000,,If 1/2 to the power of F of two\Nequals 1/2 to the power of minus
Dialogue: 0,0:32:36.50,0:32:41.38,Default,,0000,0000,0000,,one, then these powers must be\Nthe same, so F of two equals
Dialogue: 0,0:32:41.38,0:32:47.07,Default,,0000,0000,0000,,minus one. After two\Nequals minus one.
Dialogue: 0,0:32:48.69,0:32:55.88,Default,,0000,0000,0000,,Now look at\NF of four.
Dialogue: 0,0:32:55.88,0:33:01.25,Default,,0000,0000,0000,,Half of four gives us the log of\Nfour, so the base 1/2.
Dialogue: 0,0:33:02.05,0:33:07.22,Default,,0000,0000,0000,,Remember, this is equivalent to\Nsaying that one half to some
Dialogue: 0,0:33:07.22,0:33:10.51,Default,,0000,0000,0000,,power. In this case F of four.
Dialogue: 0,0:33:11.69,0:33:13.93,Default,,0000,0000,0000,,Must be equal to 4.
Dialogue: 0,0:33:14.50,0:33:18.35,Default,,0000,0000,0000,,And once again, we need to think\Nabout rewriting this right hand
Dialogue: 0,0:33:18.35,0:33:22.52,Default,,0000,0000,0000,,side to get a half so we can see\Nwhat the power is.
Dialogue: 0,0:33:22.53,0:33:25.69,Default,,0000,0000,0000,,Now for we know we can write us
Dialogue: 0,0:33:25.69,0:33:32.16,Default,,0000,0000,0000,,2 squared. And then using our\Nnegative powers we can rewrite
Dialogue: 0,0:33:32.16,0:33:34.28,Default,,0000,0000,0000,,this as one half.
Dialogue: 0,0:33:34.31,0:33:39.95,Default,,0000,0000,0000,,So the minus two, so if 1/2 to\Nthe power of F of four equals
Dialogue: 0,0:33:39.95,0:33:44.84,Default,,0000,0000,0000,,1/2 to the power of minus two,\Nthen these powers must be equal.
Dialogue: 0,0:33:44.84,0:33:47.47,Default,,0000,0000,0000,,So F of four equals minus 2.
Dialogue: 0,0:33:48.12,0:33:54.15,Default,,0000,0000,0000,,And also we want to\Nconsider some fractional
Dialogue: 0,0:33:54.15,0:34:00.18,Default,,0000,0000,0000,,arguments. So let's look at\NF of 1/2.
Dialogue: 0,0:34:00.24,0:34:02.67,Default,,0000,0000,0000,,Half of 1/2.
Dialogue: 0,0:34:02.67,0:34:08.42,Default,,0000,0000,0000,,Gives us the log of 1/2\Nto the base 1/2.
Dialogue: 0,0:34:08.92,0:34:13.36,Default,,0000,0000,0000,,So lots of haves and this is\Nequivalent to saying we've got
Dialogue: 0,0:34:13.36,0:34:18.91,Default,,0000,0000,0000,,1/2 to the power of F of 1/2.\NThat's going to be equal to 1/2.
Dialogue: 0,0:34:18.91,0:34:22.98,Default,,0000,0000,0000,,This first sight might seem a\Nlittle bit complicated, but it's
Dialogue: 0,0:34:22.98,0:34:27.79,Default,,0000,0000,0000,,not at all because 1/2 to some\Npower to give me 1/2. Well
Dialogue: 0,0:34:27.79,0:34:33.34,Default,,0000,0000,0000,,that's just half to the power of\N1, so this F of 1/2 is actually
Dialogue: 0,0:34:33.34,0:34:34.45,Default,,0000,0000,0000,,equal to 1.
Dialogue: 0,0:34:34.71,0:34:41.51,Default,,0000,0000,0000,,And finally, like\Nto consider F
Dialogue: 0,0:34:41.51,0:34:48.83,Default,,0000,0000,0000,,of 1/4. Which is\Nequal to the log of 1/4 to
Dialogue: 0,0:34:48.83,0:34:50.95,Default,,0000,0000,0000,,the base of 1/2.
Dialogue: 0,0:34:51.57,0:34:57.59,Default,,0000,0000,0000,,I remember this is equivalent to\Nwriting 1/2 to some power. In
Dialogue: 0,0:34:57.59,0:35:04.62,Default,,0000,0000,0000,,this case F of 1/4 is equal\Nto 1/4, and again, what we need
Dialogue: 0,0:35:04.62,0:35:10.14,Default,,0000,0000,0000,,to do is just think about\Nrewriting the right hand side
Dialogue: 0,0:35:10.14,0:35:15.67,Default,,0000,0000,0000,,and actually this is the same as\None over 2 squared.
Dialogue: 0,0:35:16.13,0:35:18.94,Default,,0000,0000,0000,,Which is the same as one half.
Dialogue: 0,0:35:19.53,0:35:25.60,Default,,0000,0000,0000,,Squad So 1/2\Nto some power equals 1/2
Dialogue: 0,0:35:25.60,0:35:31.96,Default,,0000,0000,0000,,squared. The powers must be\Nequal, so F of 1/4 must equal 2.
Dialogue: 0,0:35:31.96,0:35:35.24,Default,,0000,0000,0000,,After 1/4 equals 2.
Dialogue: 0,0:35:35.85,0:35:41.32,Default,,0000,0000,0000,,So as usual, put these results\Ninto a table so we can plot a
Dialogue: 0,0:35:41.32,0:35:42.89,Default,,0000,0000,0000,,graph of the function.
Dialogue: 0,0:35:43.80,0:35:48.16,Default,,0000,0000,0000,,It's just the table over here\NX&F of X.
Dialogue: 0,0:35:51.05,0:35:54.50,Default,,0000,0000,0000,,Arguments were one
Dialogue: 0,0:35:54.50,0:36:00.31,Default,,0000,0000,0000,,quarter 1/2. 1,\Ntwo and four.
Dialogue: 0,0:36:00.91,0:36:06.81,Default,,0000,0000,0000,,And the corresponding outputs.\NThere were two one 0 -
Dialogue: 0,0:36:06.81,0:36:13.89,Default,,0000,0000,0000,,1 and minus two. So let's\Nplot these points. So first of
Dialogue: 0,0:36:13.89,0:36:15.66,Default,,0000,0000,0000,,all some axes.
Dialogue: 0,0:36:16.32,0:36:17.48,Default,,0000,0000,0000,,F of X.
Dialogue: 0,0:36:18.26,0:36:25.14,Default,,0000,0000,0000,,X. Vertically, we need\Nto go from minus 2 + 2, so no
Dialogue: 0,0:36:25.14,0:36:26.80,Default,,0000,0000,0000,,problems minus 1 - 2.
Dialogue: 0,0:36:27.42,0:36:29.62,Default,,0000,0000,0000,,One and two.
Dialogue: 0,0:36:30.47,0:36:32.81,Default,,0000,0000,0000,,And horizontally we need to go\Nall the way up to four.
Dialogue: 0,0:36:33.39,0:36:36.75,Default,,0000,0000,0000,,So 1 two.
Dialogue: 0,0:36:37.27,0:36:44.20,Default,,0000,0000,0000,,314 Let's\Nplot the points 1/4 and two.
Dialogue: 0,0:36:44.73,0:36:48.81,Default,,0000,0000,0000,,1/2 and\None.
Dialogue: 0,0:36:50.77,0:36:53.08,Default,,0000,0000,0000,,One and 0.
Dialogue: 0,0:36:53.64,0:36:57.22,Default,,0000,0000,0000,,Two negative one.
Dialogue: 0,0:36:57.77,0:37:01.30,Default,,0000,0000,0000,,Four and negative
Dialogue: 0,0:37:01.30,0:37:06.87,Default,,0000,0000,0000,,2. House before going to try\Nand draw a smooth curve
Dialogue: 0,0:37:06.87,0:37:07.99,Default,,0000,0000,0000,,through these points.
Dialogue: 0,0:37:15.96,0:37:23.27,Default,,0000,0000,0000,,OK, excellent and this is F of X\Nequals the log of X to base 1/2.
Dialogue: 0,0:37:23.27,0:37:27.38,Default,,0000,0000,0000,,Actually this demonstrates the\Ngeneral shape for functions of
Dialogue: 0,0:37:27.38,0:37:34.70,Default,,0000,0000,0000,,the form F of X equals log of\NX to the base were a is equal
Dialogue: 0,0:37:34.70,0:37:37.44,Default,,0000,0000,0000,,to the number between zero and
Dialogue: 0,0:37:37.44,0:37:41.77,Default,,0000,0000,0000,,one. But what happens as a\Nvaries within those boundaries?
Dialogue: 0,0:37:42.34,0:37:46.92,Default,,0000,0000,0000,,Well, by looking at the sketch\Nof a few functions like that, we
Dialogue: 0,0:37:46.92,0:37:51.49,Default,,0000,0000,0000,,should be able to see what's\Ngoing on. So just do my axes.
Dialogue: 0,0:37:52.60,0:37:54.08,Default,,0000,0000,0000,,F of X.
Dialogue: 0,0:37:54.70,0:37:59.94,Default,,0000,0000,0000,,And X. And\Nwe've just seen.
Dialogue: 0,0:38:04.35,0:38:11.05,Default,,0000,0000,0000,,F of X equals log\Nof X. It's a base
Dialogue: 0,0:38:11.05,0:38:14.87,Default,,0000,0000,0000,,1/2. Well, we might have had.
Dialogue: 0,0:38:15.91,0:38:17.35,Default,,0000,0000,0000,,Something that looked like.
Dialogue: 0,0:38:17.93,0:38:23.83,Default,,0000,0000,0000,,This. This might have been\NF of X equals.
Dialogue: 0,0:38:23.83,0:38:28.14,Default,,0000,0000,0000,,The log of X so base one\Nover E. Remember either
Dialogue: 0,0:38:28.14,0:38:31.28,Default,,0000,0000,0000,,exponential number or we\Nmight even have hard.
Dialogue: 0,0:38:35.15,0:38:38.96,Default,,0000,0000,0000,,Something which looked like\Nthis. This might have been F of
Dialogue: 0,0:38:38.96,0:38:42.19,Default,,0000,0000,0000,,X equals. Log of X.
Dialogue: 0,0:38:42.72,0:38:48.00,Default,,0000,0000,0000,,Base 1/5\NSo what's happening for
Dialogue: 0,0:38:48.00,0:38:51.96,Default,,0000,0000,0000,,different values of a well, we\Ncan see that for the bigger
Dialogue: 0,0:38:51.96,0:38:57.27,Default,,0000,0000,0000,,values of a. The output\Ndecreases more quickly as
Dialogue: 0,0:38:57.27,0:38:59.11,Default,,0000,0000,0000,,the arguments increases.
Dialogue: 0,0:39:00.74,0:39:05.47,Default,,0000,0000,0000,,As a couple of other important\Npoints to notice here as well,
Dialogue: 0,0:39:05.47,0:39:10.98,Default,,0000,0000,0000,,firstly, is this .1 again on the\NX axis, and in fact we notice
Dialogue: 0,0:39:10.98,0:39:16.11,Default,,0000,0000,0000,,that F of one equals 0\Nregardless of our value of A and
Dialogue: 0,0:39:16.11,0:39:18.08,Default,,0000,0000,0000,,that's true for any value.
Dialogue: 0,0:39:19.06,0:39:23.11,Default,,0000,0000,0000,,Secondly, is once again this\Nthing about the positive
Dialogue: 0,0:39:23.11,0:39:27.16,Default,,0000,0000,0000,,arguments we can see\Ngraphically. Once again, the X
Dialogue: 0,0:39:27.16,0:39:29.86,Default,,0000,0000,0000,,has to be more than 0.
Dialogue: 0,0:39:30.86,0:39:35.24,Default,,0000,0000,0000,,So now we know what happens\Nfor all the different values
Dialogue: 0,0:39:35.24,0:39:38.02,Default,,0000,0000,0000,,of a when we considering\Nlogarithm functions.
Dialogue: 0,0:39:39.28,0:39:42.74,Default,,0000,0000,0000,,Once again, you may have\Nnotice some symmetry.
Dialogue: 0,0:39:43.88,0:39:49.71,Default,,0000,0000,0000,,This time the symmetry was\Ncentered on the X axis.
Dialogue: 0,0:39:51.14,0:39:55.24,Default,,0000,0000,0000,,If I actually draw two of\Nthe curves here.
Dialogue: 0,0:40:00.67,0:40:06.62,Default,,0000,0000,0000,,This one might have represented\NF of X equals the log of X to
Dialogue: 0,0:40:06.62,0:40:13.58,Default,,0000,0000,0000,,base 2. This will might\Nrepresent F of X equals log of X
Dialogue: 0,0:40:13.58,0:40:18.51,Default,,0000,0000,0000,,to the base 1/2, and then forget\Nvery important .1.
Dialogue: 0,0:40:18.92,0:40:21.07,Default,,0000,0000,0000,,And we can see here that
Dialogue: 0,0:40:21.07,0:40:26.76,Default,,0000,0000,0000,,actually. The base two function\Nis a reflection in the X axis of
Dialogue: 0,0:40:26.76,0:40:31.48,Default,,0000,0000,0000,,the function, which has a base\N1/2, and in fact that could have
Dialogue: 0,0:40:31.48,0:40:36.56,Default,,0000,0000,0000,,been five and one, 5th or E and\None over E as the base,
Dialogue: 0,0:40:36.56,0:40:43.50,Default,,0000,0000,0000,,generally speaking. F of X\Nequals the log of X to
Dialogue: 0,0:40:43.50,0:40:46.69,Default,,0000,0000,0000,,base A. Is
Dialogue: 0,0:40:46.69,0:40:49.52,Default,,0000,0000,0000,,a reflection.
Dialogue: 0,0:40:50.02,0:40:57.22,Default,,0000,0000,0000,,Of.\NF of X equals.
Dialogue: 0,0:40:57.86,0:41:03.94,Default,,0000,0000,0000,,Log of X to base one over A That\Nis in the X axis.
Dialogue: 0,0:41:04.45,0:41:10.27,Default,,0000,0000,0000,,Accent, so now we've looked at\Nreflections in the X axis in the
Dialogue: 0,0:41:10.27,0:41:15.20,Default,,0000,0000,0000,,F of X axis. We've looked at\Nexponential functions, and we've
Dialogue: 0,0:41:15.20,0:41:20.13,Default,,0000,0000,0000,,looked at logarithm functions.\NThe final thing I'd like to look
Dialogue: 0,0:41:20.13,0:41:25.51,Default,,0000,0000,0000,,at in this video is whether\Nthere is a link between these
Dialogue: 0,0:41:25.51,0:41:30.88,Default,,0000,0000,0000,,two functions. Firstly, the\Nfunction F of X equals Y to the
Dialogue: 0,0:41:30.88,0:41:32.67,Default,,0000,0000,0000,,X. Remember the exponential
Dialogue: 0,0:41:32.67,0:41:36.77,Default,,0000,0000,0000,,function. And the second\Nfunction F of X equals the
Dialogue: 0,0:41:36.77,0:41:40.14,Default,,0000,0000,0000,,natural log of X, which we\Nmentioned briefly earlier on
Dialogue: 0,0:41:40.14,0:41:43.85,Default,,0000,0000,0000,,now. Remember, this means the\Nlog of X to base A.
Dialogue: 0,0:41:45.14,0:41:49.20,Default,,0000,0000,0000,,Well, good place to start would\Nbe to look at the graphs of the
Dialogue: 0,0:41:49.20,0:41:53.91,Default,,0000,0000,0000,,functions. So I'll do that now.\NWe've got F of X.
Dialogue: 0,0:41:55.91,0:42:00.91,Default,,0000,0000,0000,,And X. And remember, F of X\Nequals E to the X.
Dialogue: 0,0:42:05.02,0:42:09.82,Default,,0000,0000,0000,,Run along like this. Is this\Nimportant? .1 F of X axis.
Dialogue: 0,0:42:10.43,0:42:13.61,Default,,0000,0000,0000,,Stuff of X equals E to the X.
Dialogue: 0,0:42:14.58,0:42:17.10,Default,,0000,0000,0000,,And if of X equals\Na natural log of X.
Dialogue: 0,0:42:18.30,0:42:19.55,Default,,0000,0000,0000,,Came from here.
Dialogue: 0,0:42:21.14,0:42:25.49,Default,,0000,0000,0000,,I'm run along something like\Nthis once again going through
Dialogue: 0,0:42:25.49,0:42:28.54,Default,,0000,0000,0000,,that important .1 on the X axis.
Dialogue: 0,0:42:28.55,0:42:30.54,Default,,0000,0000,0000,,Sex equals and natural log of X.
Dialogue: 0,0:42:31.42,0:42:37.37,Default,,0000,0000,0000,,Now instead of link, I think\Nhelpful line to draw in here is
Dialogue: 0,0:42:37.37,0:42:38.75,Default,,0000,0000,0000,,this dotted line.
Dialogue: 0,0:42:40.76,0:42:42.11,Default,,0000,0000,0000,,This dotted line.
Dialogue: 0,0:42:43.23,0:42:48.57,Default,,0000,0000,0000,,Represents the graph of the\Nlinear function F of X equals X.
Dialogue: 0,0:42:49.48,0:42:54.65,Default,,0000,0000,0000,,I want to put that in. We can\Nsee almost immediately that the
Dialogue: 0,0:42:54.65,0:42:57.84,Default,,0000,0000,0000,,exponential function is a\Nreflection of the natural
Dialogue: 0,0:42:57.84,0:43:03.01,Default,,0000,0000,0000,,logarithm function in the line F\Nof X equals X. What does that
Dialogue: 0,0:43:03.01,0:43:07.39,Default,,0000,0000,0000,,mean? Well, this is equivalent\Nto saying that the axes have
Dialogue: 0,0:43:07.39,0:43:11.77,Default,,0000,0000,0000,,been swapped around, so to move\Nfrom this function to this
Dialogue: 0,0:43:11.77,0:43:14.16,Default,,0000,0000,0000,,function, all my ex file use.
Dialogue: 0,0:43:14.17,0:43:19.63,Default,,0000,0000,0000,,Have gone to become F of X\Nvalues and all my F of X
Dialogue: 0,0:43:19.63,0:43:23.53,Default,,0000,0000,0000,,values have gone to become\NX values. In other words,
Dialogue: 0,0:43:23.53,0:43:26.65,Default,,0000,0000,0000,,the inputs and outputs\Nhave been swapped around.
Dialogue: 0,0:43:27.95,0:43:30.15,Default,,0000,0000,0000,,So In summary.
Dialogue: 0,0:43:31.20,0:43:37.96,Default,,0000,0000,0000,,This means that the function F\Nof X equals E to the X.
Dialogue: 0,0:43:38.42,0:43:41.46,Default,,0000,0000,0000,,Is the inverse?
Dialogue: 0,0:43:42.30,0:43:48.50,Default,,0000,0000,0000,,Of the function F\Nof X equals a
Dialogue: 0,0:43:48.50,0:43:51.60,Default,,0000,0000,0000,,natural log of X.