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