0:00:01.040,0:00:04.730 In this video, we'll be looking[br]at exponential functions and 0:00:04.730,0:00:08.420 logarithm functions, and I'd[br]like to start off by thinking 0:00:08.420,0:00:13.955 about functions of the form F of[br]X equals A to the power of X, 0:00:13.955,0:00:17.645 where a is representing real[br]positive numbers. I'm going to 0:00:17.645,0:00:22.442 split this up into three cases.[br]First of all, the case when a 0:00:22.442,0:00:27.608 equals 1 hour, then going to[br]look at the case when A is more 0:00:27.608,0:00:31.298 than one, and finally I'll look[br]at the case where. 0:00:31.310,0:00:36.539 Is between zero and one. So[br]first of all. 0:00:36.790,0:00:42.880 When a equals 1, this will give[br]us the function F of X equals 1 0:00:42.880,0:00:48.970 to the power of X and we can see[br]that once the power of anything 0:00:48.970,0:00:54.654 is actually one. So this is the[br]linear function F of X equals 1. 0:00:55.300,0:01:00.300 So that's quite straightforward,[br]and Secondly, I'd like to look 0:01:00.300,0:01:04.300 at the case where a is more than 0:01:04.300,0:01:09.040 one. And to demonstrate what[br]happens in this case, I'd like 0:01:09.040,0:01:13.240 to consider a specific example.[br]In this case, I'll choose A to 0:01:13.240,0:01:18.490 be equal to two, which gives us[br]the function F of X equals 2 to 0:01:18.490,0:01:19.890 the power of X. 0:01:20.620,0:01:24.130 Now a good place to start with[br]these kind of functions is to 0:01:24.130,0:01:26.560 look at for some different[br]values of the argument. 0:01:27.320,0:01:32.435 So starts off by looking at F of[br]0, which is actually equal to 2 0:01:32.435,0:01:33.799 to the power 0. 0:01:34.330,0:01:38.543 And we know that anything to the[br]power 0 equals 1. 0:01:38.660,0:01:41.929 Next, we'll look at F of one. 0:01:41.930,0:01:44.178 Which is 2 to the power of 1. 0:01:44.980,0:01:47.460 And two to the power of one is 0:01:47.460,0:01:51.455 2. And we can look at F of 0:01:51.455,0:01:58.870 two. Which is 2 squared,[br]which is 4. So quite 0:01:58.870,0:02:01.670 straightforward, and finally F 0:02:01.670,0:02:07.940 of three. Which is 2 to the[br]power of three which actually 0:02:07.940,0:02:09.398 gives us 8. 0:02:09.410,0:02:13.780 Also want to consider some[br]negative arguments as well, so 0:02:13.780,0:02:17.276 if we look at F of minus one. 0:02:17.290,0:02:21.526 This is 2 to the power of[br]minus one. And remember when 0:02:21.526,0:02:25.409 we have a negative power,[br]that means that we have to 0:02:25.409,0:02:28.939 invert our number so we[br]actually end up with one 0:02:28.939,0:02:29.292 half. 0:02:30.430,0:02:37.646 If you look at F of minus two[br]I guess is 2 to the power of 0:02:37.646,0:02:42.156 minus two. Once again, this[br]negative power makes we've got 0:02:42.156,0:02:48.019 one over 2 squared and it's 2[br]squared is 4's actually gives us 0:02:48.019,0:02:54.060 1/4. And final arguments are[br]consider is F of minus three. 0:02:54.790,0:03:00.394 Which is 2 to the power of[br]minus three, which gives us 0:03:00.394,0:03:06.465 one over 2 cubed and two[br]cubes 8. So we get one 8th. 0:03:07.510,0:03:11.129 I'm going to take these[br]results now and put them into 0:03:11.129,0:03:15.735 a table and we can use that[br]table to help us plot a graph 0:03:15.735,0:03:16.722 of the function. 0:03:18.760,0:03:23.593 So our table, the[br]values of X&F of X. 0:03:25.430,0:03:33.026 We just come from minus 3[br]- 2 - 1 zero, 1 0:03:33.026,0:03:36.824 two and three and the value 0:03:36.824,0:03:38.090 hardware 1/8. 0:03:38.590,0:03:45.064 1/4[br]1/2 one, 2, 0:03:45.064,0:03:48.658 four and eight. 0:03:49.190,0:03:53.555 So we're going to plot these now[br]so we can get a graph of the 0:03:53.555,0:03:57.530 function. So do RF of X 0:03:57.530,0:04:03.784 axis here. And X axis[br]horizontally. So on the X axis 0:04:03.784,0:04:07.186 we need to go from minus three 0:04:07.186,0:04:14.926 to +3. So minus 1[br]- 2 - 3 and 0:04:14.926,0:04:18.490 one. Two and three this 0:04:18.490,0:04:23.276 way. And on the vertical scale,[br]the F of X axis, we need to go 0:04:23.276,0:04:28.972 up to 8. So[br]12345678. Make 0:04:28.972,0:04:31.680 sure we 0:04:31.680,0:04:38.855 label that.[br]So now let's plot the 0:04:38.855,0:04:39.646 points. 0:04:41.040,0:04:43.008 Minus 3 1/8. 0:04:43.860,0:04:46.786 Minus 2 0:04:46.786,0:04:54.160 1/4. Minus[br]1 1/2. 0:04:55.170,0:04:56.700 Zero and one. 0:04:57.630,0:05:05.040 12[br]214 0:05:06.660,0:05:10.300 And finally, three and eight. 0:05:11.950,0:05:17.438 And so we need to try and draw a[br]smooth curve through the points. 0:05:22.240,0:05:29.040 And this is the graph[br]of the function F of 0:05:29.040,0:05:34.480 X equals 2 to the[br]power of X. 0:05:35.600,0:05:39.480 Now this is actually quite[br]clearly shows the general shape 0:05:39.480,0:05:45.688 of graphs of the functions where[br]F of X equals A to the X, and a 0:05:45.688,0:05:50.344 is more than one. However, what[br]happens when we vary the value 0:05:50.344,0:05:56.025 of A? Well, by looking at a few[br]sketches of a few different 0:05:56.025,0:05:57.565 graphs, they should become 0:05:57.565,0:06:01.110 clear. So I have my axes again. 0:06:01.810,0:06:05.530 F of X. 0:06:06.310,0:06:13.272 X horizontally.[br]We've just spotted the graph of 0:06:13.272,0:06:19.699 this function. Which one through[br]1 F of X axis and this was F of 0:06:19.699,0:06:22.523 X equals 2 to the power of X. 0:06:23.040,0:06:26.708 If we were to look at this. 0:06:27.470,0:06:28.220 Graph. 0:06:29.600,0:06:36.958 This might represent F of X[br]equals 5 to the power of X. 0:06:38.750,0:06:40.556 If I was look at this graph. 0:06:43.330,0:06:49.050 This might represent the graph[br]of the function F of X equals 10 0:06:49.050,0:06:51.250 to the power of X. 0:06:52.120,0:06:55.600 So what's actually happening[br]here? Well, for bigger values of 0:06:55.600,0:07:01.881 a. We can see that the output[br]increases more quickly as the 0:07:01.881,0:07:02.843 arguments increases. 0:07:04.150,0:07:08.110 Another couple of important[br]points to notice here are first 0:07:08.110,0:07:12.466 of all that every single graph[br]that I've sketched here comes 0:07:12.466,0:07:17.218 through this .0 one, and in[br]fact, regardless of our value of 0:07:17.218,0:07:20.610 A. F of 0. 0:07:21.370,0:07:24.916 Will equal 1 for every single[br]value of a. 0:07:25.680,0:07:32.892 Secondly, we notice that F of[br]X is always more than 0. 0:07:33.790,0:07:37.830 In other words, are output for[br]this function is always 0:07:37.830,0:07:41.093 positive. As a very[br]important feature of these 0:07:41.093,0:07:42.020 kind of functions. 0:07:43.870,0:07:46.600 The last case I would like to 0:07:46.600,0:07:53.144 consider. Is the case where a is[br]between zero and one case where 0:07:53.144,0:07:55.886 a is between zero and one. 0:07:56.490,0:08:01.494 To demonstrate this case, I[br]would like it to look at a 0:08:01.494,0:08:06.498 specific example. In this case I[br]will choose a equals 1/2, so 0:08:06.498,0:08:10.668 this means I'm looking at the[br]function F of X. 0:08:11.210,0:08:15.900 Equals 1/2 to[br]the power of X. 0:08:16.990,0:08:20.842 Now with the last example, a[br]good place to start is by 0:08:20.842,0:08:22.447 looking at some different values 0:08:22.447,0:08:27.453 for the arguments. So let's[br]first of all consider F of 0. 0:08:28.690,0:08:33.200 This will give us 1/2 to the[br]power of 0. 0:08:33.840,0:08:37.168 And as we said before, anything[br]to the power of 0 is one. 0:08:37.990,0:08:41.854 Secondly, we look at F of one. 0:08:42.580,0:08:46.260 Which is 1/2 to the power of 1. 0:08:47.040,0:08:48.610 Which is equal to 1/2. 0:08:51.020,0:08:53.318 Half of 2. 0:08:53.830,0:08:57.559 Equals 1/2 squared. 0:08:58.340,0:09:01.772 So on the top that just gives[br]US1 squared, which is one. 0:09:02.450,0:09:08.274 On the bottom 2 squared, which[br]is 4. So we end up with one 0:09:08.274,0:09:11.500 quarter. An F of three. 0:09:12.330,0:09:15.890 Is equal to 1/2 cubed. 0:09:17.150,0:09:22.120 So on the top we get one cubed[br]which just gives US1 and under 0:09:22.120,0:09:27.090 bottom 2 cubed which gives us 8.[br]So we end up with one 8th. 0:09:27.670,0:09:31.330 And as before, we also need[br]to consider some negative 0:09:31.330,0:09:31.696 arguments. 0:09:32.750,0:09:35.580 So half of minus one. 0:09:36.130,0:09:39.889 Gives us 1/2 to the power of 0:09:39.889,0:09:44.788 minus one. And remember the[br]minus sign on the power actually 0:09:44.788,0:09:49.436 inverts are fraction, so we end[br]up with two over 1 to the power 0:09:49.436,0:09:51.428 of 1, which is just two. 0:09:51.970,0:09:55.080 F of 0:09:55.080,0:10:02.424 minus 2. Gives[br]us 1/2 to the power of minus 0:10:02.424,0:10:09.510 2. Which is 2 over 1 squared.[br]You can see on the top we get 2 0:10:09.510,0:10:14.450 squared which is 4 and on the[br]bottom we just get one. So 0:10:14.450,0:10:16.350 that's actually equal to 4. 0:10:17.630,0:10:20.906 And finally, F of minus three. 0:10:21.650,0:10:25.346 Is one half to the power of 0:10:25.346,0:10:31.603 -3? Which means we get two over[br]one and we deal with the minus 0:10:31.603,0:10:36.280 sign cubed. So 2 cubed in the[br]top which is 8 and again just 0:10:36.280,0:10:39.305 the one on the bottom. So that[br]just gives us 8. 0:10:40.660,0:10:43.660 So once again, we're going to[br]take these results and put them 0:10:43.660,0:10:47.160 into a table so we can plot a[br]graph. A graph of the function. 0:10:48.540,0:10:55.428 So X&F of X[br]again for our table. 0:10:56.110,0:11:00.820 And we have values of the[br]argument ranging from minus 0:11:00.820,0:11:04.588 three, all the way up to[br]three again. 0:11:06.850,0:11:12.835 And this time the[br]values where 8421. 0:11:13.510,0:11:17.068 1/2 one quarter. 0:11:17.690,0:11:19.259 And one 8th. 0:11:20.260,0:11:23.076 So let's plot this now[br]on a graph. 0:11:24.870,0:11:31.464 So vertically we get F of X and[br]a horizontal axis. We've got X 0:11:31.464,0:11:35.703 and we're going from minus three[br]to three again. 0:11:37.830,0:11:41.274 So minus 1 - 2 - 0:11:41.274,0:11:44.200 3. 123 0:11:44.200,0:11:51.355 here. And[br]then we're going up 0:11:51.355,0:11:57.634 to 8 on the[br]vertical axis, 12345678. 0:11:57.940,0:12:03.508 So let's[br]plot the 0:12:03.508,0:12:08.276 points. First point[br]is minus three 8. 0:12:10.280,0:12:13.250 Which is around about here. 0:12:14.020,0:12:16.160 Minus two and four. 0:12:18.820,0:12:22.008 But here. Minus one and two. 0:12:23.530,0:12:26.440 But the. Zero and one. 0:12:27.140,0:12:35.228 One 1/2.[br]2 one quarter. 0:12:35.990,0:12:39.518 And finally 3 1/8. 0:12:40.070,0:12:44.893 As with our previous example, we[br]need to try and draw a smooth 0:12:44.893,0:12:46.377 curve through the points. 0:12:47.610,0:12:48.530 So. 0:12:56.870,0:13:03.910 And this represents the function[br]F of X equals 1/2 to 0:13:03.910,0:13:06.470 the power of X. 0:13:07.080,0:13:10.653 And actually this demonstrates[br]the general shape for functions 0:13:10.653,0:13:17.402 of the form F of X equals A to[br]the X when A is between zero and 0:13:17.402,0:13:21.769 one. But what happens when we[br]vary a within those boundaries? 0:13:22.380,0:13:25.148 Well, sketching a few[br]graphs of this function 0:13:25.148,0:13:26.878 will help us to see. 0:13:28.200,0:13:31.090 Just do some axes again. 0:13:31.710,0:13:33.366 So we've got F of X. 0:13:36.120,0:13:39.309 And X. We've just seen. 0:13:44.030,0:13:46.110 This curve, which was. 0:13:46.970,0:13:52.720 F of X equals 1/2 to[br]the power of X. 0:13:53.600,0:13:55.928 We might have seen. 0:14:02.120,0:14:04.348 This would have represented. 0:14:06.220,0:14:08.560 F of X equals. 0:14:09.400,0:14:14.284 1/5 to the power of X or we may[br]even have seen. 0:14:15.160,0:14:16.459 Something like this? 0:14:21.300,0:14:27.204 Which would have represented[br]maybe F of X equals 110th to the 0:14:27.204,0:14:28.680 power of X. 0:14:30.020,0:14:36.712 So we can see that for bigger[br]values of a this is. We come 0:14:36.712,0:14:41.492 down here. The output decreases[br]more slowly as the arguments 0:14:41.492,0:14:46.560 increases. And a few important[br]points to notice here. First of 0:14:46.560,0:14:50.619 all, as with the previous[br]example, F of 0 equals 1 0:14:50.619,0:14:55.785 regardless of the value of A and[br]in fact, as long as a is 0:14:55.785,0:14:59.106 positive and real, this will[br]always be the case. 0:15:00.900,0:15:05.674 Second thing to notice, as with[br]the previous example, is that 0:15:05.674,0:15:10.882 our output F of X is always[br]positive, so output is always 0:15:10.882,0:15:12.184 more than 0. 0:15:15.020,0:15:19.376 So now we've looked at what[br]happens for all the different 0:15:19.376,0:15:22.148 values of a when A is positive 0:15:22.148,0:15:24.740 and real. No one[br]interesting thing you 0:15:24.740,0:15:27.604 might have noticed is[br]this. We've got some 0:15:27.604,0:15:31.184 symmetry going on here. If[br]I actually just put some 0:15:31.184,0:15:32.616 axes down again here. 0:15:33.790,0:15:37.758 So F of X find X you'll[br]remember. 0:15:39.600,0:15:40.968 This curve here. 0:15:42.510,0:15:47.844 Passing through the point one[br]was F of X equals 2 to the power 0:15:47.844,0:15:52.660 of X. And also[br]this curve here. 0:15:56.170,0:16:02.913 Was F of X equals 1/2[br]to the power of X? 0:16:03.900,0:16:08.954 And we can see this link it[br]'cause one of these graphs is a 0:16:08.954,0:16:14.008 reflection of the other in the F[br]of X axis. And in fact this 0:16:14.008,0:16:19.784 could have been 1/5 to the X and[br]this could have been 5 to the X 0:16:19.784,0:16:20.867 'cause generally speaking. 0:16:20.870,0:16:23.630 F of X equal to. 0:16:24.230,0:16:25.980 AX. 0:16:27.040,0:16:29.732 Is a 0:16:29.732,0:16:36.995 reflection. Of F of[br]X equals 1 over 0:16:36.995,0:16:43.755 8 to the X[br]and that is in 0:16:43.755,0:16:47.135 the F of X 0:16:47.135,0:16:54.794 axis. And one over 8 to the[br]X can also be written as A to 0:16:54.794,0:16:56.192 the minus X. 0:16:57.380,0:16:59.900 So that's an interesting point 0:16:59.900,0:17:06.180 to note. Now you might recall[br]from Chapter 2.3 that the 0:17:06.180,0:17:07.620 exponential number E. 0:17:08.310,0:17:11.350 Is approximately 0:17:11.350,0:17:16.485 2.718. Which means it falls[br]into the first category, where 0:17:16.485,0:17:20.625 a is more than one. If we're[br]going to consider the function 0:17:20.625,0:17:25.110 F of X equals E to the X, which[br]is the exponential function. 0:17:27.650,0:17:30.338 So if I do some axes again. 0:17:30.910,0:17:32.798 F of X vertically. 0:17:34.060,0:17:35.968 An ex horizontally. 0:17:39.340,0:17:40.579 This graph here. 0:17:41.730,0:17:47.205 Might represent F of X equals E[br]to the X and this is what the 0:17:47.205,0:17:48.300 exponential function actually 0:17:48.300,0:17:53.464 looks like. I might also like to[br]consider the function F of X 0:17:53.464,0:17:55.528 equals E to the minus X. 0:17:56.190,0:17:57.578 As we've just discovered. 0:17:59.620,0:18:06.097 That is a reflection in the F of[br]X axis, so this will be F of X 0:18:06.097,0:18:10.669 equals E to the minus X. And[br]remember this important .1 on 0:18:10.669,0:18:12.574 the F of X axis. 0:18:13.780,0:18:17.218 So look that exponential[br]functions and we've looked the 0:18:17.218,0:18:19.892 functions of the form A to the 0:18:19.892,0:18:22.880 X. At my work to consider 0:18:22.880,0:18:26.835 logarithm functions. And[br]logarithm functions take the 0:18:26.835,0:18:32.932 form F of X equals the log[br]of X. So particular base. In 0:18:32.932,0:18:34.339 this case a. 0:18:35.690,0:18:39.298 And as with the previous[br]example, I'd like to split my 0:18:39.298,0:18:42.906 analysis of this into three[br]parts. First of all, looking at 0:18:42.906,0:18:44.218 when a is one. 0:18:44.890,0:18:48.934 Second of all, looking at when[br]there is more than one, and 0:18:48.934,0:18:52.978 finally when a is a number[br]between zero and one, and as 0:18:52.978,0:18:57.022 with the previous example, a is[br]only going to be positive real 0:18:57.022,0:19:01.990 numbers. So first of all, what[br]happens when a equals 1? 0:19:02.970,0:19:08.378 Well, this means we'll get a[br]function F of X equals the log 0:19:08.378,0:19:10.458 of X to base one. 0:19:11.040,0:19:14.008 Remember, this is equivalent. 0:19:14.930,0:19:16.700 To say that this number one. 0:19:17.220,0:19:20.010 To some power. 0:19:20.050,0:19:26.030 F of X is equal to X, just[br]like earlier on, this is 0:19:26.030,0:19:32.658 equivalent. Generally. 2A[br]to the power of F of X 0:19:32.658,0:19:33.660 equals X. 0:19:35.180,0:19:40.262 So when we look at this, we can[br]see that we can only get 0:19:40.262,0:19:43.529 solutions when we consider the[br]arguments X equals 1. 0:19:44.130,0:19:48.666 And in fact, if we look at the[br]arguments X equals 1, there is 0:19:48.666,0:19:52.878 an infinite number of answers be[br]cause one to any power will give 0:19:52.878,0:19:56.766 us one. So actually this is not[br]a valid function because we've 0:19:56.766,0:20:00.330 got many outputs for just one[br]single input. So that's what 0:20:00.330,0:20:04.542 happens for a equals 1 would[br]happen for a is more than one. 0:20:05.240,0:20:09.332 Let's look at the case when I is[br]more than one. 0:20:09.870,0:20:15.044 So this means we're looking at[br]the function F of X equals and 0:20:15.044,0:20:19.820 in this case I will choose a[br]equals 2 again to demonstrate 0:20:19.820,0:20:25.930 what's happening. So F of X[br]equals the log of X to base 2. 0:20:26.610,0:20:29.354 And remember, this is 0:20:29.354,0:20:33.950 equivalent. The same 2 to the[br]power of F of X. 0:20:34.450,0:20:40.030 Equals X. So as with all[br]functions, when we get to this 0:20:40.030,0:20:43.286 kind of situation, we want to[br]start looking at some different 0:20:43.286,0:20:47.134 values for the argument to help[br]us plot a graph of the function. 0:20:47.820,0:20:51.356 However, before we do that, we[br]might just want to take a closer 0:20:51.356,0:20:52.988 look at what's going on here. 0:20:54.290,0:20:59.990 Because 2 to the power of F of[br]X. In other words, two to some 0:20:59.990,0:21:04.930 power can never ever be negative[br]or 0. This is a positive number. 0:21:04.930,0:21:08.730 Whatever we choose, and since[br]that's more than zero, and 0:21:08.730,0:21:13.670 that's equal to X, this means[br]that X must be more than 0. 0:21:14.230,0:21:16.814 And X represents our[br]arguments, which means I'm 0:21:16.814,0:21:19.721 only going to look at[br]positive arguments for this 0:21:19.721,0:21:20.044 reason. 0:21:21.460,0:21:25.267 So start off by looking[br]at F of one. 0:21:26.470,0:21:30.058 Now F of one gives us. 0:21:30.140,0:21:33.318 The log of one to base 2. 0:21:34.480,0:21:37.888 North actually means remember. 0:21:38.440,0:21:44.824 Is that two to some power? Half[br]of one is equal to 1, so two to 0:21:44.824,0:21:50.410 what power will give me one?[br]Well, it must be 2 to the power 0:21:50.410,0:21:54.799 zero. Give me one because[br]anything to the power zero gives 0:21:54.799,0:22:00.784 me one. So this half of one must[br]be 0. So therefore F of one 0:22:00.784,0:22:07.197 equals 0. Next I look[br]at F of two. 0:22:07.990,0:22:10.751 And this is means that we've got[br]the log of two. 0:22:11.540,0:22:12.620 To base 2. 0:22:13.220,0:22:17.059 And remember that this is[br]equivalent to saying that two to 0:22:17.059,0:22:22.294 some power. In this case F of[br]two is equal to two SO2 to what 0:22:22.294,0:22:24.039 power will give Me 2? 0:22:24.540,0:22:29.835 Well, we know that 2 to the[br]power of one will give Me 2, so 0:22:29.835,0:22:32.659 this F of two must be equal to 0:22:32.659,0:22:39.820 1. And also going to look[br]at F of F of four. 0:22:40.430,0:22:47.306 Now F of four means that[br]we've got the log of four 0:22:47.306,0:22:50.744 to base two. Remember, this is 0:22:50.744,0:22:53.028 equivalent. Just saying that 0:22:53.028,0:22:57.594 we've got. Two to some[br]power. In this case, F 0:22:57.594,0:23:00.012 of four is equal to 4. 0:23:01.120,0:23:03.880 Now 2 to what power will give me 0:23:03.880,0:23:09.532 4? Well, it's actually going to[br]be 2 squared, so it must be 2 to 0:23:09.532,0:23:13.984 the power of two. Will give me[br]4, so therefore we can see that 0:23:13.984,0:23:16.210 F of four will actually give Me 0:23:16.210,0:23:21.843 2. I also want to consider[br]some fractional arguments here 0:23:21.843,0:23:23.911 between zero and one. 0:23:24.650,0:23:27.744 So if I look at for example. 0:23:28.410,0:23:30.849 F of 1/2. 0:23:31.630,0:23:37.714 OK, this means we've got the log[br]of 1/2 to the base 2. 0:23:38.420,0:23:41.396 And this is equivalent[br]to saying. 0:23:42.580,0:23:49.626 The two to some power. In this[br]case F of 1/2 equals 1/2. 0:23:50.810,0:23:53.966 Now this time it's a little bit[br]more tricky to see actually 0:23:53.966,0:23:55.018 what's going on here. 0:23:55.810,0:24:02.124 But remember, we can write 1/2[br]as 2 to the power of minus one. 0:24:02.850,0:24:06.030 Because remember about that[br]minus power that we talked about 0:24:06.030,0:24:10.482 earlier on. So 2 to the power of[br]sampling equals 2 to the power 0:24:10.482,0:24:13.344 of minus one. This something[br]must be minus one. 0:24:13.980,0:24:20.710 So therefore. Half[br]of 1/2 actually equals minus 0:24:20.710,0:24:28.163 one. And the final argument[br]I want to consider is F of 0:24:28.163,0:24:34.494 1/4. And this gives us[br]the log of 1/4. 0:24:34.540,0:24:35.660 So the base 2. 0:24:36.160,0:24:39.928 So this is equivalent to saying. 0:24:40.600,0:24:48.328 That two to some power. In this[br]case F of 1/4 is equal to 0:24:48.328,0:24:52.586 1/4. And once again, it's not an[br]easy step just to see exactly 0:24:52.586,0:24:56.282 what's going on here. Straight[br]away. We want to try and rewrite 0:24:56.282,0:24:57.514 this right hand side. 0:24:58.040,0:25:03.738 Now one over 4 is the same as[br]one over 2 squared. +2 squared 0:25:03.738,0:25:09.029 is just the same as four, and[br]remember about the minus sign so 0:25:09.029,0:25:15.541 we can put that onto the top and[br]we get 2 to the power of minus 0:25:15.541,0:25:21.239 two. So here 2 to the power of[br]something equals 2 to the power 0:25:21.239,0:25:25.716 of minus two. That something[br]must be minus two, so therefore 0:25:25.716,0:25:28.158 F of 1/4 equals minus 2. 0:25:29.210,0:25:33.084 So what we're going to do now?[br]We're going to put these results 0:25:33.084,0:25:36.660 into a table so we can plot a[br]graph of the function. 0:25:37.520,0:25:43.556 So we've got[br]X&F of X. 0:25:44.420,0:25:50.948 Now X values we[br]chose. We had one 0:25:50.948,0:25:54.140 quarter. We had one half. 0:25:54.930,0:25:58.790 1, two and four. 0:25:59.300,0:26:05.340 And the corresponding outputs[br]Here were minus 2 - 1 0:26:05.340,0:26:07.756 zero one and two. 0:26:08.490,0:26:12.282 So let's look at plotting the[br]graph of this function now, so 0:26:12.282,0:26:14.178 as before, we want some axes 0:26:14.178,0:26:17.802 here. So F of 0:26:17.802,0:26:21.366 X vertically. And X 0:26:21.366,0:26:28.036 horizontally. Horizontally we[br]need to go from 0:26:28.036,0:26:31.600 one 3:45, so just 0:26:31.600,0:26:37.611 go 1234. Includes everything we[br]need and vertically we need to 0:26:37.611,0:26:40.425 go from minus 2 + 2. 0:26:40.520,0:26:42.520 So minus 1 - 2. 0:26:43.160,0:26:48.326 I'm one and two, so let's[br]put the points. 0:26:49.160,0:26:56.996 1/4 and minus 2. First of[br]all, 1/4 minus two 1/2 - 0:26:56.996,0:27:03.898 1. So it's 1/2[br]- 1 one and 0. 0:27:04.860,0:27:06.040 21 0:27:07.240,0:27:11.070 And finally, four and two. 0:27:11.940,0:27:15.672 So now we want to try and[br]draw a smooth curve through 0:27:15.672,0:27:19.093 the point so we can see the[br]graph of the function. 0:27:20.340,0:27:27.990 Excellent, so this[br]is F of 0:27:27.990,0:27:35.640 X equals the[br]log of X. 0:27:35.640,0:27:39.760 These two. And actually, this[br]represents the general shape for 0:27:39.760,0:27:44.785 functions of the form F of X[br]equals the log of X to base a 0:27:44.785,0:27:49.140 when A is more than one. But[br]what happens as we very well, 0:27:49.140,0:27:53.495 let's have a look at a few[br]sketches of some graphs of some 0:27:53.495,0:27:56.845 different functions and that[br]should help us to see what's 0:27:56.845,0:28:02.590 going on. So if I have[br]F of X vertically. 0:28:03.410,0:28:06.959 And X horizontally. 0:28:07.920,0:28:09.540 We've just seen. 0:28:11.280,0:28:14.754 F of 0:28:14.754,0:28:20.948 X equals. Log[br]of X to base 2. 0:28:22.470,0:28:24.150 And if I was to draw this. 0:28:27.300,0:28:28.869 This might represent. 0:28:29.570,0:28:36.245 F of X equals the log of X[br]to the base E. Remember E being 0:28:36.245,0:28:39.805 the number 2.718, the[br]exponential number and actually 0:28:39.805,0:28:44.255 this is called the natural log[br]and is sometimes written. 0:28:44.900,0:28:46.010 LNX 0:28:47.320,0:28:49.740 And finally, it might have. 0:28:53.150,0:28:56.810 F of X equals. 0:28:57.360,0:29:02.443 Log of AXA Base 5 maybe so we[br]can see that what's happening 0:29:02.443,0:29:04.789 here for bigger values of a. 0:29:05.910,0:29:09.335 The output is increasing more 0:29:09.335,0:29:11.935 slowly. As the arguments 0:29:11.935,0:29:17.360 increases. Now a few important[br]points to notice here, the first 0:29:17.360,0:29:22.002 one. It's a notice this point[br]here, this one on the X axis. 0:29:22.860,0:29:25.548 Regardless of our value of A. 0:29:26.100,0:29:32.988 F of one will always be 0.[br]That's true for all values of a 0:29:32.988,0:29:36.771 here. Second thing to notice is[br]something we touched upon 0:29:36.771,0:29:39.661 earlier on is just point about[br]the arguments always being 0:29:39.661,0:29:42.840 positive and we can see this[br]graphically. Here we see we've 0:29:42.840,0:29:46.308 got no points to the left of the[br]F of X axis. 0:29:47.020,0:29:50.870 And so X is always more than 0:29:50.870,0:29:55.730 0. The final case[br]I want to look at. 0:29:56.930,0:30:02.832 Is the case worth a is between[br]zero and one. And to demonstrate 0:30:02.832,0:30:08.280 this case I will look at the[br]specific example where a equals 0:30:08.280,0:30:14.048 1/2. And if I equals 1/2, the[br]function we're going to be 0:30:14.048,0:30:20.468 looking at is F of X equals the[br]log of X to the base 1/2. 0:30:21.410,0:30:27.108 So remember we can rewrite this.[br]This is equivalent to saying 0:30:27.108,0:30:34.360 that one half to some power. In[br]this case F of X is equal 0:30:34.360,0:30:38.718 to X. Answer the previous[br]example. We need to think 0:30:38.718,0:30:41.562 carefully about which arguments[br]we're going to consider now. 0:30:42.100,0:30:47.655 Because 1/2 to any power will[br]always give me a positive 0:30:47.655,0:30:52.705 number. In other words, 1/2. So[br]the F of X. 0:30:53.220,0:30:54.940 Is always more than 0. 0:30:55.720,0:30:59.528 And since this is equal to X,[br]this means that X is more than 0:30:59.528,0:31:02.590 0. And so many going to consider 0:31:02.590,0:31:07.984 positive arguments. So first of[br]all, I can set up F of one. 0:31:08.750,0:31:15.133 Half of 1 means that we've[br]got the log of one to the 0:31:15.133,0:31:16.606 base of 1/2. 0:31:17.780,0:31:19.600 Remember, this is equivalent. 0:31:20.650,0:31:28.110 Just saying.[br]1/2 to some power. In this case, 0:31:28.110,0:31:31.610 F of one is equal to 1. 0:31:33.770,0:31:38.114 So how does this work? 1/2 to[br]some power equals 1. Remember 0:31:38.114,0:31:42.458 anything to the power of 0[br]equals 1, so F of one. 0:31:43.000,0:31:46.138 Must be 0. 0:31:46.140,0:31:52.029 Secondly,[br]F of two. 0:31:52.970,0:31:55.385 This gives us the log of two. 0:31:55.890,0:31:57.678 So the base 1/2. 0:31:58.370,0:32:00.475 Remember, this is equivalent to 0:32:00.475,0:32:06.612 saying. That we've got 1/2 to[br]some power. In this case F of 0:32:06.612,0:32:10.428 two. Is equal to 2. 0:32:11.120,0:32:15.380 So how do we find out what this[br]powers got to be? 0:32:15.920,0:32:19.968 Well, we want to look at[br]rewriting this number 2, and 0:32:19.968,0:32:23.648 this is where the minus negative[br]powers come in useful. 0:32:24.390,0:32:29.822 So we can actually write this as[br]1/2 to the power of minus one. 0:32:30.500,0:32:36.500 If 1/2 to the power of F of two[br]equals 1/2 to the power of minus 0:32:36.500,0:32:41.375 one, then these powers must be[br]the same, so F of two equals 0:32:41.375,0:32:47.070 minus one. After two[br]equals minus one. 0:32:48.690,0:32:55.878 Now look at[br]F of four. 0:32:55.880,0:33:01.249 Half of four gives us the log of[br]four, so the base 1/2. 0:33:02.050,0:33:07.220 Remember, this is equivalent to[br]saying that one half to some 0:33:07.220,0:33:10.510 power. In this case F of four. 0:33:11.690,0:33:13.930 Must be equal to 4. 0:33:14.500,0:33:18.352 And once again, we need to think[br]about rewriting this right hand 0:33:18.352,0:33:22.525 side to get a half so we can see[br]what the power is. 0:33:22.530,0:33:25.690 Now for we know we can write us 0:33:25.690,0:33:32.159 2 squared. And then using our[br]negative powers we can rewrite 0:33:32.159,0:33:34.283 this as one half. 0:33:34.310,0:33:39.950 So the minus two, so if 1/2 to[br]the power of F of four equals 0:33:39.950,0:33:44.838 1/2 to the power of minus two,[br]then these powers must be equal. 0:33:44.838,0:33:47.470 So F of four equals minus 2. 0:33:48.120,0:33:54.152 And also we want to[br]consider some fractional 0:33:54.152,0:34:00.184 arguments. So let's look at[br]F of 1/2. 0:34:00.240,0:34:02.670 Half of 1/2. 0:34:02.670,0:34:08.420 Gives us the log of 1/2[br]to the base 1/2. 0:34:08.920,0:34:13.360 So lots of haves and this is[br]equivalent to saying we've got 0:34:13.360,0:34:18.910 1/2 to the power of F of 1/2.[br]That's going to be equal to 1/2. 0:34:18.910,0:34:22.980 This first sight might seem a[br]little bit complicated, but it's 0:34:22.980,0:34:27.790 not at all because 1/2 to some[br]power to give me 1/2. Well 0:34:27.790,0:34:33.340 that's just half to the power of[br]1, so this F of 1/2 is actually 0:34:33.340,0:34:34.450 equal to 1. 0:34:34.710,0:34:41.508 And finally, like[br]to consider F 0:34:41.508,0:34:48.830 of 1/4. Which is[br]equal to the log of 1/4 to 0:34:48.830,0:34:50.950 the base of 1/2. 0:34:51.570,0:34:57.594 I remember this is equivalent to[br]writing 1/2 to some power. In 0:34:57.594,0:35:04.622 this case F of 1/4 is equal[br]to 1/4, and again, what we need 0:35:04.622,0:35:10.144 to do is just think about[br]rewriting the right hand side 0:35:10.144,0:35:15.666 and actually this is the same as[br]one over 2 squared. 0:35:16.130,0:35:18.937 Which is the same as one half. 0:35:19.530,0:35:25.603 Squad So 1/2[br]to some power equals 1/2 0:35:25.603,0:35:31.960 squared. The powers must be[br]equal, so F of 1/4 must equal 2. 0:35:31.960,0:35:35.240 After 1/4 equals 2. 0:35:35.850,0:35:41.324 So as usual, put these results[br]into a table so we can plot a 0:35:41.324,0:35:42.888 graph of the function. 0:35:43.800,0:35:48.165 It's just the table over here[br]X&F of X. 0:35:51.050,0:35:54.500 Arguments were one 0:35:54.500,0:36:00.308 quarter 1/2. 1,[br]two and four. 0:36:00.910,0:36:06.810 And the corresponding outputs.[br]There were two one 0 - 0:36:06.810,0:36:13.890 1 and minus two. So let's[br]plot these points. So first of 0:36:13.890,0:36:15.660 all some axes. 0:36:16.320,0:36:17.478 F of X. 0:36:18.260,0:36:25.142 X. Vertically, we need[br]to go from minus 2 + 2, so no 0:36:25.142,0:36:26.797 problems minus 1 - 2. 0:36:27.420,0:36:29.619 One and two. 0:36:30.470,0:36:32.810 And horizontally we need to go[br]all the way up to four. 0:36:33.390,0:36:36.750 So 1 two. 0:36:37.270,0:36:44.198 314 Let's[br]plot the points 1/4 and two. 0:36:44.730,0:36:48.810 1/2 and[br]one. 0:36:50.770,0:36:53.080 One and 0. 0:36:53.640,0:36:57.219 Two negative one. 0:36:57.770,0:37:01.301 Four and negative 0:37:01.301,0:37:06.870 2. House before going to try[br]and draw a smooth curve 0:37:06.870,0:37:07.992 through these points. 0:37:15.960,0:37:23.272 OK, excellent and this is F of X[br]equals the log of X to base 1/2. 0:37:23.272,0:37:27.385 Actually this demonstrates the[br]general shape for functions of 0:37:27.385,0:37:34.697 the form F of X equals log of[br]X to the base were a is equal 0:37:34.697,0:37:37.439 to the number between zero and 0:37:37.439,0:37:41.768 one. But what happens as a[br]varies within those boundaries? 0:37:42.340,0:37:46.916 Well, by looking at the sketch[br]of a few functions like that, we 0:37:46.916,0:37:51.492 should be able to see what's[br]going on. So just do my axes. 0:37:52.600,0:37:54.079 F of X. 0:37:54.700,0:37:59.940 And X. And[br]we've just seen. 0:38:04.350,0:38:11.050 F of X equals log[br]of X. It's a base 0:38:11.050,0:38:14.870 1/2. Well, we might have had. 0:38:15.910,0:38:17.350 Something that looked like. 0:38:17.930,0:38:23.828 This. This might have been[br]F of X equals. 0:38:23.830,0:38:28.142 The log of X so base one[br]over E. Remember either 0:38:28.142,0:38:31.278 exponential number or we[br]might even have hard. 0:38:35.150,0:38:38.956 Something which looked like[br]this. This might have been F of 0:38:38.956,0:38:42.188 X equals. Log of X. 0:38:42.720,0:38:48.000 Base 1/5[br]So what's happening for 0:38:48.000,0:38:51.960 different values of a well, we[br]can see that for the bigger 0:38:51.960,0:38:57.272 values of a. The output[br]decreases more quickly as 0:38:57.272,0:38:59.108 the arguments increases. 0:39:00.740,0:39:05.468 As a couple of other important[br]points to notice here as well, 0:39:05.468,0:39:10.984 firstly, is this .1 again on the[br]X axis, and in fact we notice 0:39:10.984,0:39:16.106 that F of one equals 0[br]regardless of our value of A and 0:39:16.106,0:39:18.076 that's true for any value. 0:39:19.060,0:39:23.110 Secondly, is once again this[br]thing about the positive 0:39:23.110,0:39:27.160 arguments we can see[br]graphically. Once again, the X 0:39:27.160,0:39:29.860 has to be more than 0. 0:39:30.860,0:39:35.238 So now we know what happens[br]for all the different values 0:39:35.238,0:39:38.024 of a when we considering[br]logarithm functions. 0:39:39.280,0:39:42.736 Once again, you may have[br]notice some symmetry. 0:39:43.880,0:39:49.710 This time the symmetry was[br]centered on the X axis. 0:39:51.140,0:39:55.244 If I actually draw two of[br]the curves here. 0:40:00.670,0:40:06.620 This one might have represented[br]F of X equals the log of X to 0:40:06.620,0:40:13.583 base 2. This will might[br]represent F of X equals log of X 0:40:13.583,0:40:18.513 to the base 1/2, and then forget[br]very important .1. 0:40:18.920,0:40:21.068 And we can see here that 0:40:21.068,0:40:26.756 actually. The base two function[br]is a reflection in the X axis of 0:40:26.756,0:40:31.475 the function, which has a base[br]1/2, and in fact that could have 0:40:31.475,0:40:36.557 been five and one, 5th or E and[br]one over E as the base, 0:40:36.557,0:40:43.505 generally speaking. F of X[br]equals the log of X to 0:40:43.505,0:40:46.693 base A. Is 0:40:46.693,0:40:49.519 a reflection. 0:40:50.020,0:40:57.220 Of.[br]F of X equals. 0:40:57.860,0:41:03.936 Log of X to base one over A That[br]is in the X axis. 0:41:04.450,0:41:10.274 Accent, so now we've looked at[br]reflections in the X axis in the 0:41:10.274,0:41:15.202 F of X axis. We've looked at[br]exponential functions, and we've 0:41:15.202,0:41:20.130 looked at logarithm functions.[br]The final thing I'd like to look 0:41:20.130,0:41:25.506 at in this video is whether[br]there is a link between these 0:41:25.506,0:41:30.882 two functions. Firstly, the[br]function F of X equals Y to the 0:41:30.882,0:41:32.674 X. Remember the exponential 0:41:32.674,0:41:36.773 function. And the second[br]function F of X equals the 0:41:36.773,0:41:40.143 natural log of X, which we[br]mentioned briefly earlier on 0:41:40.143,0:41:43.850 now. Remember, this means the[br]log of X to base A. 0:41:45.140,0:41:49.200 Well, good place to start would[br]be to look at the graphs of the 0:41:49.200,0:41:53.910 functions. So I'll do that now.[br]We've got F of X. 0:41:55.910,0:42:00.910 And X. And remember, F of X[br]equals E to the X. 0:42:05.020,0:42:09.820 Run along like this. Is this[br]important? .1 F of X axis. 0:42:10.430,0:42:13.606 Stuff of X equals E to the X. 0:42:14.580,0:42:17.100 And if of X equals[br]a natural log of X. 0:42:18.300,0:42:19.548 Came from here. 0:42:21.140,0:42:25.490 I'm run along something like[br]this once again going through 0:42:25.490,0:42:28.535 that important .1 on the X axis. 0:42:28.550,0:42:30.538 Sex equals and natural log of X. 0:42:31.420,0:42:37.374 Now instead of link, I think[br]helpful line to draw in here is 0:42:37.374,0:42:38.748 this dotted line. 0:42:40.760,0:42:42.110 This dotted line. 0:42:43.230,0:42:48.570 Represents the graph of the[br]linear function F of X equals X. 0:42:49.480,0:42:54.654 I want to put that in. We can[br]see almost immediately that the 0:42:54.654,0:42:57.838 exponential function is a[br]reflection of the natural 0:42:57.838,0:43:03.012 logarithm function in the line F[br]of X equals X. What does that 0:43:03.012,0:43:07.390 mean? Well, this is equivalent[br]to saying that the axes have 0:43:07.390,0:43:11.768 been swapped around, so to move[br]from this function to this 0:43:11.768,0:43:14.156 function, all my ex file use. 0:43:14.170,0:43:19.630 Have gone to become F of X[br]values and all my F of X 0:43:19.630,0:43:23.530 values have gone to become[br]X values. In other words, 0:43:23.530,0:43:26.650 the inputs and outputs[br]have been swapped around. 0:43:27.950,0:43:30.149 So In summary. 0:43:31.200,0:43:37.960 This means that the function F[br]of X equals E to the X. 0:43:38.420,0:43:41.459 Is the inverse? 0:43:42.300,0:43:48.500 Of the function F[br]of X equals a 0:43:48.500,0:43:51.600 natural log of X.