WEBVTT 00:00:00.760 --> 00:00:04.951 In this video, we're going to be looking at logarithms, but 00:00:04.951 --> 00:00:09.142 before we can deal with logarithms, what we need to have 00:00:09.142 --> 00:00:11.428 a look at our indices 16. 00:00:12.460 --> 00:00:19.164 We know that we can write 16 as a power of two, so this is 2. 00:00:19.800 --> 00:00:22.660 Raised to the power 4. 00:00:23.240 --> 00:00:27.354 Now need to give some names to some of these numbers. 00:00:28.660 --> 00:00:31.130 So this number here 4. 00:00:32.810 --> 00:00:37.240 Gets called a number of different things. I just called 00:00:37.240 --> 00:00:38.569 it a power. 00:00:40.010 --> 00:00:43.190 Sometimes it gets called. 00:00:43.940 --> 00:00:45.200 Exponent 00:00:46.290 --> 00:00:52.786 Sometimes. Index. So that's the four. What about 00:00:52.786 --> 00:00:56.818 this number? This number doesn't usually get a name, but it does 00:00:56.818 --> 00:01:02.856 have one. It's referred to as the base. 00:01:03.630 --> 00:01:06.970 So another example might be 00:01:06.970 --> 00:01:12.858 64. 64 is 8 to the power 2. 00:01:13.890 --> 00:01:20.622 So in this case two is the power, the exponent or 00:01:20.622 --> 00:01:25.518 the index, and here 8 is the base. 00:01:26.830 --> 00:01:33.188 So having got the language clear for indices or powers, what 00:01:33.188 --> 00:01:35.600 about logarithms? Well. 00:01:36.220 --> 00:01:41.841 Let's see what the motivation might be for wanting to use 00:01:41.841 --> 00:01:42.352 logarithms. 00:01:43.080 --> 00:01:49.488 No. As we've just 00:01:49.488 --> 00:01:53.590 seen, 16 is 2 raised to the 00:01:53.590 --> 00:01:55.348 power of 4. 00:01:56.950 --> 00:02:03.882 8. Similarly, is 2 raised to the power of 3? 00:02:05.360 --> 00:02:12.341 Now, supposing we want it to multiply 16 by 8, well, one way 00:02:12.341 --> 00:02:15.563 of doing it would be long 00:02:15.563 --> 00:02:20.637 multiplication. We can do that and the answer is 128. 00:02:21.970 --> 00:02:26.007 That is another way rather than having to go through this 00:02:26.007 --> 00:02:29.677 multiplication. Some which if the numbers were other than 16 00:02:29.677 --> 00:02:34.815 and eight would be very long. Can we not make use of the way 00:02:34.815 --> 00:02:35.916 we've expressed these? 00:02:37.000 --> 00:02:40.380 16 times by 8. 00:02:41.110 --> 00:02:47.610 Is 2 to the power four times by 2 to the power 3. 00:02:48.110 --> 00:02:53.310 From laws of indices, we know that when we do this kind of 00:02:53.310 --> 00:02:57.710 calculation, what we do is we simply add the indices together. 00:02:58.960 --> 00:03:02.950 So what was a multiplication? Some we've 00:03:02.950 --> 00:03:05.800 reduced to an addition some. 00:03:07.300 --> 00:03:11.720 Similarly, we could do this sort of thing with division, 00:03:11.720 --> 00:03:19.234 so if we have 16 / 8, that would be 2 to the power, 4 / 2 00:03:19.234 --> 00:03:24.538 to the power three, and that would just be too, because we 00:03:24.538 --> 00:03:26.306 would subtract the indices. 00:03:27.350 --> 00:03:31.221 Now, if we had a table of 00:03:31.221 --> 00:03:38.159 numbers. Where we listed these powers then all we would need to 00:03:38.159 --> 00:03:40.754 do is look them up. 00:03:41.380 --> 00:03:47.908 Do addition. And then look back again at what this 2 to the 00:03:47.908 --> 00:03:50.596 power 7 actually meant that it 00:03:50.596 --> 00:03:57.744 means 128. Now this idea is the whole basis of 00:03:57.744 --> 00:04:03.336 logarithms. And they were devised in the late 16th 00:04:03.336 --> 00:04:07.592 century by two mathematicians working in dependently John 00:04:07.592 --> 00:04:09.720 Napier and Henry Briggs. 00:04:11.160 --> 00:04:14.592 So what exactly is 00:04:14.592 --> 00:04:19.290 a logarithm? Let's start with this 16 00:04:19.290 --> 00:04:22.866 is 2 to the power 4. 00:04:24.280 --> 00:04:29.259 This is the power index or the exponent, and this is the base. 00:04:30.610 --> 00:04:38.338 If we take the logarithm, which we usually write as log to 00:04:38.338 --> 00:04:45.568 base 2. Of this number 16, then the logarithm is 00:04:45.568 --> 00:04:50.510 4. This power index or exponent becomes 00:04:50.510 --> 00:04:56.090 the logarithm, so we take logs to a base. 00:04:58.280 --> 00:05:04.820 And it's the base that gets raised to a particular power in 00:05:04.820 --> 00:05:10.815 indices, and that power becomes the logarithm. So these two are 00:05:10.815 --> 00:05:15.175 equivalent statements. If we write one, we automatically 00:05:15.175 --> 00:05:16.810 imply the other. 00:05:17.480 --> 00:05:24.676 So if I say that 64 is 8 to the power, two, that is 00:05:24.676 --> 00:05:29.816 exactly the same as saying that the log of 64. 00:05:30.380 --> 00:05:36.032 Two base eight is 2. These two statements are exactly the same. 00:05:36.032 --> 00:05:42.155 So if I write a statement down this side, is the exactly the 00:05:42.155 --> 00:05:46.865 same statement written down here? They both mean the same 00:05:46.865 --> 00:05:53.930 thing, so this side I say the log to base three of 27 is 3. 00:05:53.930 --> 00:06:00.995 what I am saying is that 3 to the power three is equal to 27. 00:06:02.590 --> 00:06:07.642 So these statements down the left here or exactly the same as 00:06:07.642 --> 00:06:09.747 these statements down the right. 00:06:10.830 --> 00:06:16.344 So. We can write this down as a general 00:06:16.344 --> 00:06:19.348 statement. If we have a number X. 00:06:20.460 --> 00:06:24.436 And we can write X as a 00:06:24.436 --> 00:06:29.676 number A. Raised to the power and then the equivalent 00:06:29.676 --> 00:06:33.564 statement in logarithms is to say that the log. 00:06:34.300 --> 00:06:41.040 To the base a of X is equal to N. 00:06:41.040 --> 00:06:43.736 These two are equivalent 00:06:43.736 --> 00:06:51.130 statements. Let's just develop a little bit of that, supposing X 00:06:51.130 --> 00:06:53.554 were equal to 10. 00:06:55.100 --> 00:07:00.721 Then we can write 10 as 10 to the power one. 00:07:01.860 --> 00:07:08.430 So if I now take the log to base 9:50. 00:07:09.280 --> 00:07:14.320 Then be'cause 10 can be written as 10 to the power one. 00:07:14.850 --> 00:07:20.908 Then it's logarithm to the base. 10 must be one. Similarly, if I 00:07:20.908 --> 00:07:23.704 had, X is equal to 2. 00:07:24.290 --> 00:07:27.962 Then two is just 2 to the power one. 00:07:29.140 --> 00:07:31.254 And so if I take the log. 00:07:31.900 --> 00:07:37.345 To base two off to again, that's just one. 00:07:38.380 --> 00:07:43.616 So let's make that general. If X were equal to a, then we know 00:07:43.616 --> 00:07:47.730 that we can write that as A to the power one. 00:07:48.530 --> 00:07:56.090 And so that the log to base a of a is there for one. 00:07:56.930 --> 00:08:03.818 And this gives us a general rule that works for any of our base 00:08:03.818 --> 00:08:09.664 numbers a. Let's see if we can generate some laws of 00:08:09.664 --> 00:08:15.256 logarithms. We know that there are laws of indices. Can we have 00:08:15.256 --> 00:08:20.382 similar laws of logarithms, and in one respect we've already had 00:08:20.382 --> 00:08:21.780 the first law? 00:08:21.780 --> 00:08:28.828 So. Let's have a look. Let's take 2 numbers X&Y and what we 00:08:28.828 --> 00:08:34.816 want to be able to do is multiply X&Y together. As a 00:08:34.816 --> 00:08:40.804 general thing, so we say X is A to the power N. 00:08:41.320 --> 00:08:45.100 And why is A to the power 00:08:45.100 --> 00:08:51.776 N? Now this statement X is A to the power, N is the 00:08:51.776 --> 00:08:58.118 equivalent of saying the log of X to base a is equal to N. 00:08:58.620 --> 00:09:04.847 This statement is the equivalent of saying the log of Y also to 00:09:04.847 --> 00:09:07.721 base a is equal to M. 00:09:08.730 --> 00:09:15.880 We want to multiply these two together so X times by Y is 00:09:15.880 --> 00:09:22.480 A to the power N times by A to the power M. 00:09:23.730 --> 00:09:30.290 And what do we get here? A to the power N times by 8 to the 00:09:30.290 --> 00:09:35.620 power N means that we add the indices together. So that's A to 00:09:35.620 --> 00:09:37.260 the N Plus M. 00:09:38.350 --> 00:09:41.520 So what is the logarithm? 00:09:42.150 --> 00:09:45.250 Two base a of XY. 00:09:45.920 --> 00:09:49.264 While quite clearly from the definition we've had. 00:09:49.820 --> 00:09:57.317 X times by Y is A to the power N plus M, so the logarithm is N 00:09:57.317 --> 00:10:05.096 plus N. But N we know is the log of X to base a log 00:10:05.096 --> 00:10:12.968 of X to base A and similarly M is the log of Y to base A. 00:10:13.020 --> 00:10:20.980 And so there we have our first law 00:10:20.980 --> 00:10:26.642 of logarithms. That if we want to multiply 2 numbers 00:10:26.642 --> 00:10:31.699 together, we get the log of the product by adding the logs of 00:10:31.699 --> 00:10:33.255 the two individual numbers. 00:10:34.180 --> 00:10:40.280 That's our first law. What about our second law again? 00:10:40.790 --> 00:10:47.225 Let's start with X can be written as A to the power N. 00:10:47.750 --> 00:10:50.648 Now, what if we take X? 00:10:51.340 --> 00:10:54.484 And we raise it to the power M. 00:10:55.690 --> 00:10:56.959 That would be. 00:10:58.230 --> 00:11:03.130 A to the N all raised to the power M. 00:11:04.190 --> 00:11:09.594 And by our laws of indices, we know that in order to do that, 00:11:09.594 --> 00:11:14.612 we multiply the N and the M together, giving us NN as the 00:11:14.612 --> 00:11:15.770 power of A. 00:11:16.510 --> 00:11:22.919 So what's the logarithm here? Log of X to the power M to 00:11:22.919 --> 00:11:24.398 the base A. 00:11:25.740 --> 00:11:32.120 Equals well, by definition, that must be this number here. End 00:11:32.120 --> 00:11:33.860 times by N. 00:11:34.770 --> 00:11:36.998 An times by N. 00:11:37.580 --> 00:11:44.570 Equals. Well, this statement here it's 00:11:44.570 --> 00:11:50.216 equivalent statement. Is that the log of X? 00:11:50.790 --> 00:11:57.328 To base a is equal to N, so instead of NI can write this. 00:11:58.330 --> 00:12:05.185 And I must multiply it by NM times the log of X to base a. 00:12:05.185 --> 00:12:11.126 There we have our second law of logarithms that if we want to 00:12:11.126 --> 00:12:14.325 raise a number to a given power. 00:12:15.700 --> 00:12:20.458 Then we can do that by taking the log, multiplying that log by 00:12:20.458 --> 00:12:25.582 M. That will give us the log of the number to the given power, 00:12:25.582 --> 00:12:30.706 and then we can look it up in reverse. So our second law of 00:12:30.706 --> 00:12:35.968 logarithms. Our third law, well, in a way we've already met with 00:12:35.968 --> 00:12:39.948 third law because now we've done multiplication of two different 00:12:39.948 --> 00:12:43.530 numbers. Repeated multiplication of the same number. So what 00:12:43.530 --> 00:12:46.316 we're going to look at now is 00:12:46.316 --> 00:12:52.730 division. So again, will define X to be A to the power N. 00:12:53.600 --> 00:12:57.407 And why to be A to the power M? 00:12:57.960 --> 00:13:05.032 Will write down what that means in terms of logarithms. Log of X 00:13:05.032 --> 00:13:08.296 to the base A is N. 00:13:08.860 --> 00:13:15.470 The log of Y to the base A is N. 00:13:16.500 --> 00:13:22.356 And we're going to have a look at X divided by Y. 00:13:23.640 --> 00:13:31.624 So that's A to the N divided by A to the N and our laws of 00:13:31.624 --> 00:13:37.113 indices tell us that we do this calculation by subtracting these 00:13:37.113 --> 00:13:40.606 indices, so that's A to the N 00:13:40.606 --> 00:13:47.762 minus M. So what's the log of this quantity log to 00:13:47.762 --> 00:13:54.698 base a of X divided by Y equals or by definition it 00:13:54.698 --> 00:13:57.588 must be this index here. 00:13:58.290 --> 00:14:01.299 An minus M. 00:14:02.380 --> 00:14:08.310 And N. Is this quantity the log to base a of X? 00:14:08.880 --> 00:14:12.820 Log to base A 4X. 00:14:13.620 --> 00:14:16.777 And this quantity M is up here. 00:14:17.510 --> 00:14:24.504 The log. Of Y two base A and. So there 00:14:24.504 --> 00:14:27.942 is our third law of logarithms. 00:14:29.050 --> 00:14:32.344 That if we want the log of a quotient. 00:14:33.480 --> 00:14:38.310 The log of a division some. Then we subtract the logs of the two 00:14:38.310 --> 00:14:40.035 numbers that were working with. 00:14:41.350 --> 00:14:44.076 1 final 00:14:44.076 --> 00:14:47.888 point. Final general point. 00:14:49.260 --> 00:14:55.620 What happens if we have A to the power 0? 00:14:56.620 --> 00:15:01.633 We know that anything raised to the power zero 00:15:01.633 --> 00:15:03.861 from indices is one. 00:15:06.290 --> 00:15:09.600 Well, what does that mean? 00:15:10.240 --> 00:15:11.950 About the log. 00:15:12.770 --> 00:15:16.194 To base A of 00:15:16.194 --> 00:15:23.332 one. Well, since we can write 1 as A to the power zero, that 00:15:23.332 --> 00:15:28.420 means that the log of one to base a must be 0. 00:15:29.200 --> 00:15:33.281 And so the log of one in any base is 0. 00:15:34.020 --> 00:15:39.792 Now that does make sense. Let's just think about it. If you were 00:15:39.792 --> 00:15:45.337 doing multiplication. One times by 6 is 6 * 1 doesn't affect the 00:15:45.337 --> 00:15:51.369 six, so if we were to do the same in logs, we want the log of 00:15:51.369 --> 00:15:55.893 one plus the log of six. Wouldn't want to change the log 00:15:55.893 --> 00:16:00.794 of six simply because we knew we were multiplying by one, so it 00:16:00.794 --> 00:16:04.564 makes sense for the log of one to be 0. 00:16:05.710 --> 00:16:11.760 Let's have a look now at some more examples of calculating 00:16:11.760 --> 00:16:17.260 logarithms or various numbers to particular basis. So let's take 00:16:17.260 --> 00:16:21.110 the log of 512 to base 2. 00:16:21.970 --> 00:16:29.894 Well, this is the same as asking what's 512 as a power of two 00:16:29.894 --> 00:16:36.120 512 equals 2 to what power? What's that power up there? 00:16:36.120 --> 00:16:41.780 Well, 512 is in fact 2 to the power 9. 00:16:42.580 --> 00:16:44.520 And so by definition. 00:16:45.110 --> 00:16:48.876 The log of 512 to base 2. 00:16:49.400 --> 00:16:50.990 Is 9. 00:16:52.780 --> 00:16:54.608 What about the log? 00:16:55.450 --> 00:17:01.064 To base eight of one over 64. 00:17:01.740 --> 00:17:09.432 This is the same as asking what is one over 64 as 00:17:09.432 --> 00:17:13.980 a power. Of eight. 00:17:15.910 --> 00:17:20.740 Well, what is that? We know that one over 64. 00:17:21.290 --> 00:17:23.610 Is 64. 00:17:25.260 --> 00:17:28.520 To the minus one. 00:17:30.720 --> 00:17:38.154 We also know that 64 is 8 squared, 8 times by 8 and so 00:17:38.154 --> 00:17:45.057 that tells us that one over 64 is 8 tool, the minus 2. 00:17:45.590 --> 00:17:52.220 And so the logarithm of one over 64 to base eight is minus. 00:17:52.780 --> 00:17:53.360 2. 00:17:55.580 --> 00:17:58.732 What about? The log 00:17:58.732 --> 00:18:05.530 of. 25 Tool base 5. 00:18:06.950 --> 00:18:14.030 Well, this is the same as asking what is 25 written as 00:18:14.030 --> 00:18:16.390 5 to the power. 00:18:16.910 --> 00:18:20.846 What's that power that index? What goes there? 00:18:21.390 --> 00:18:28.395 Well, what we do know is that it's 5 squared and so the log of 00:18:28.395 --> 00:18:31.664 25 two base five is just two. 00:18:32.870 --> 00:18:37.886 I want to do another calculation which is going to look very 00:18:37.886 --> 00:18:42.902 similar to this one and I'll need to compare it with this 00:18:42.902 --> 00:18:47.082 calculation, so I'm going to repeat this statement over the 00:18:47.082 --> 00:18:50.660 page. So we've got log. 00:18:51.260 --> 00:18:57.716 Of 25 to base five, we know that 00:18:57.716 --> 00:19:03.967 that's two. Now what if I interchange the number and the 00:19:03.967 --> 00:19:11.270 base? So that I'm asking the question, what's the log of 00:19:11.270 --> 00:19:13.746 five to base 25? 00:19:14.630 --> 00:19:16.586 Now that's the same as asking. 00:19:17.980 --> 00:19:19.620 If I have 5. 00:19:20.170 --> 00:19:26.506 How can I write it as a power of 25 Watts that 00:19:26.506 --> 00:19:27.562 index there? 00:19:28.730 --> 00:19:35.002 Well, one of the things I do know is that the square root of 00:19:35.002 --> 00:19:40.826 25 is 5 and I can write a square root as a power. 00:19:41.660 --> 00:19:42.660 1/2 00:19:43.730 --> 00:19:48.278 So what that tells me is that the log of five to 00:19:48.278 --> 00:19:49.794 base 25 is 1/2. 00:19:51.000 --> 00:19:56.126 And what seems to have happened here is that by interchanging 00:19:56.126 --> 00:19:58.456 the number with the base. 00:19:59.560 --> 00:20:02.680 We got one over the logarithm. 00:20:04.230 --> 00:20:07.118 Let's check that by looking at another example. 00:20:09.050 --> 00:20:14.966 We know that eight can be written as 2 to the power 00:20:14.966 --> 00:20:19.896 three, and of course that means that the log of 00:20:19.896 --> 00:20:22.854 eight to base two is 3. 00:20:24.040 --> 00:20:30.774 But what if we interchange the number with the base? So we ask 00:20:30.774 --> 00:20:35.954 ourselves what's the log of two to the base 8? 00:20:36.800 --> 00:20:41.354 And that's the equivalent of saying to ourselves. How can I 00:20:41.354 --> 00:20:46.322 write 2 as a power of eight? What goes there? What's that 00:20:46.322 --> 00:20:53.787 index? Well, two is the cube root of 00:20:53.787 --> 00:20:59.960 8. Which we write like that. But another way of writing the cube 00:20:59.960 --> 00:21:03.740 root is to write it as 8 to the 00:21:03.740 --> 00:21:10.768 power 1/3. That tells us that the log of two to base eight 00:21:10.768 --> 00:21:16.840 is 1/3. And so we see again, that by interchanging the number 00:21:16.840 --> 00:21:18.094 and the base. 00:21:18.670 --> 00:21:21.970 We get one over the original 00:21:21.970 --> 00:21:27.400 logarithm. And that's true in general. I haven't proved it by 00:21:27.400 --> 00:21:31.888 showing you those two examples, but I have been able to 00:21:31.888 --> 00:21:37.192 demonstrate it and that is true in general that if we have the 00:21:37.192 --> 00:21:39.640 log of B to base A. 00:21:40.280 --> 00:21:46.013 Then that is equal to one over the log 00:21:46.013 --> 00:21:49.198 of A to base be. 00:21:52.620 --> 00:21:55.572 Now. We've done a lot of work 00:21:55.572 --> 00:22:00.708 with different bases. So the question that we might ask is, 00:22:00.708 --> 00:22:04.767 are there any standard basis? Are logarhythms calculated using 00:22:04.767 --> 00:22:08.826 particular basis? And of course the answer is yes. 00:22:08.840 --> 00:22:13.784 What are those bases? Well, one of the common. The two common 00:22:13.784 --> 00:22:19.140 ones is 10. If you look on your Calculator, you will see a 00:22:19.140 --> 00:22:20.788 button that's labeled log. 00:22:21.460 --> 00:22:28.793 Just log. That button that's labeled log gives you logs to 00:22:28.793 --> 00:22:33.876 base 10. So for instance, if you put in 100. 00:22:34.390 --> 00:22:39.577 And press the log key. It will give you the answer to because 00:22:39.577 --> 00:22:42.370 the log of 100 to base 10. 00:22:43.240 --> 00:22:48.514 That's the log of 10 squared to base. 10 00:22:48.514 --> 00:22:50.858 clearly gives us 2. 00:22:52.050 --> 00:22:55.690 What's the other common base? Well, the other common bases 00:22:55.690 --> 00:23:01.126 base E. To letter, not a number you might say, but 00:23:01.126 --> 00:23:06.094 remember pie is also a letter, a Greek letter, and the number 00:23:06.094 --> 00:23:10.648 and E and π share something in common. They both have 00:23:10.648 --> 00:23:14.788 infinite decimal expansions, so E is a very special number. 00:23:16.370 --> 00:23:22.694 How big is it? Well to three decimal places? It's 2.718, but 00:23:22.694 --> 00:23:27.964 that's the three decimal places. Remember, it's got an infinite 00:23:27.964 --> 00:23:32.766 decimal expansion. So if you look on your Calculator, you 00:23:32.766 --> 00:23:35.440 will see a button that's got Ln 00:23:35.440 --> 00:23:43.064 on it. And that button Ln stands for logs to base E. So if you 00:23:43.064 --> 00:23:50.552 put the number 3 in it and press Ln, it will give you the log to 00:23:50.552 --> 00:23:52.424 base E of three. 00:23:53.770 --> 00:23:56.660 These logs have a name. 00:23:57.160 --> 00:24:03.224 Sometimes they're called Napier Ian Logarhythms After John 00:24:03.224 --> 00:24:08.530 Napier, and sometimes they're called natural logarithms. 00:24:10.970 --> 00:24:16.085 Logs to base E of the logs that get used in calculus. So it is 00:24:16.085 --> 00:24:19.836 important to know about logs because they're going to come up 00:24:19.836 --> 00:24:23.587 as a regular part of the calculus, and in particular when 00:24:23.587 --> 00:24:24.610 we're solving differential 00:24:24.610 --> 00:24:30.841 equations. So let's see if we can develop a way of using 00:24:30.841 --> 00:24:33.727 logarithms and see a use for 00:24:33.727 --> 00:24:38.550 them. This isn't going to be a using terms of calculus, but in 00:24:38.550 --> 00:24:40.100 terms of solving a particular 00:24:40.100 --> 00:24:46.810 kind of equation. So supposing we've got 3 to the power 00:24:46.810 --> 00:24:48.700 X equals 5. 00:24:49.780 --> 00:24:54.020 How would we solve that equation? The unknown is up 00:24:54.020 --> 00:24:56.988 here, it's an index, it's an exponent. 00:24:58.530 --> 00:25:03.704 We need to get it down out of being an exponent down to 00:25:03.704 --> 00:25:08.082 being an ordinary number and one of the ways of doing 00:25:08.082 --> 00:25:12.858 that is to take logarithms, so I'm going to take the log 00:25:12.858 --> 00:25:15.246 of both sides to base 10. 00:25:16.590 --> 00:25:18.725 So I'm going to have the log. 00:25:19.720 --> 00:25:26.286 Of three to the power, X is equal to the log of five. Now 00:25:26.286 --> 00:25:30.976 notice I haven't written the 10 in on the base. 00:25:31.710 --> 00:25:36.160 And that's because I'm using the convention that I've just 00:25:36.160 --> 00:25:40.165 expressed that log now means log to base 10. 00:25:41.450 --> 00:25:46.784 Well, I've got the log of three raised to the power X and of 00:25:46.784 --> 00:25:52.499 course one of the things that I do know from my laws of logs is 00:25:52.499 --> 00:25:57.833 that I'm raising fun raising to the power X. Then what I need to 00:25:57.833 --> 00:26:00.119 do is multiply the logarithm by 00:26:00.119 --> 00:26:07.068 X. Well now log of three and log of five aren't just 00:26:07.068 --> 00:26:08.652 numbers, nothing more. 00:26:09.400 --> 00:26:15.991 So in the same way that I might solve an equation such as 00:26:15.991 --> 00:26:21.061 three X equals 12 by dividing both sides by three. 00:26:21.070 --> 00:26:28.252 Then I'm going to do exactly the same with this, and I'm going to 00:26:28.252 --> 00:26:31.843 divide both sides by the log of 00:26:31.843 --> 00:26:37.261 three. And that calculation can be finished off using a 00:26:37.261 --> 00:26:42.728 Calculator. We take the log of 5 divided by the log of three 00:26:42.728 --> 00:26:45.869 using our Calculator and get the answer for X. 00:26:46.710 --> 00:26:48.852 Let's just have a look at one 00:26:48.852 --> 00:26:55.714 more example. If we have three to the X is equal to 5 to the 00:26:55.714 --> 00:27:01.018 X minus two. What then again, the unknown is X and again it's 00:27:01.018 --> 00:27:06.322 upstairs, so to speak. It's in the index in the power and we 00:27:06.322 --> 00:27:11.218 need to bring it back downstairs to the normal level. So again 00:27:11.218 --> 00:27:16.930 will take logs of both sides, log of three to the power X is 00:27:16.930 --> 00:27:20.194 equal to the log of 5 to the 00:27:20.194 --> 00:27:25.000 power. Minus two in the same way, if we're raising to the 00:27:25.000 --> 00:27:29.440 power, then in order to get the logarithm we need to multiply. 00:27:30.190 --> 00:27:37.390 The log of three by X and the log of five by 00:27:37.390 --> 00:27:39.190 X minus 2. 00:27:40.360 --> 00:27:44.529 Well, this is now just an ordinary equation log of three 00:27:44.529 --> 00:27:49.077 and log of five and nothing more than numbers. So let's multiply 00:27:49.077 --> 00:27:50.972 out this bracket this side. 00:27:51.680 --> 00:27:59.480 So we're going to have X log 5 - 2 00:27:59.480 --> 00:28:05.900 log 5. Let's gather together some terms in X and move this as 00:28:05.900 --> 00:28:10.970 a number over to this site so it's positive. So we're going to 00:28:10.970 --> 00:28:15.650 add this to both sides, which will give Me 2 log 5. 00:28:16.680 --> 00:28:24.646 And I'm going to take this away from both sides, so X log 5 00:28:24.646 --> 00:28:26.922 minus X log 3. 00:28:27.990 --> 00:28:33.315 I've got all my ex is together here, so I'm going to take X out 00:28:33.315 --> 00:28:35.090 as a common factor 2. 00:28:35.780 --> 00:28:43.280 Log 5 is X times log 5 minus log 3 and close the bracket. So 00:28:43.280 --> 00:28:50.780 now to get X all I need to do is to divide both sides by 00:28:50.780 --> 00:28:57.780 this number here log 5 minus log 3 because that's all it is. It's 00:28:57.780 --> 00:29:04.280 just a number, so let's do that right. The line down again to 00:29:04.280 --> 00:29:05.780 log 5 is. 00:29:05.790 --> 00:29:12.882 Equal to X times log 5 minus log 3. 00:29:14.920 --> 00:29:21.856 And I'm going to divide both sides by this number 2 log 00:29:21.856 --> 00:29:28.792 5 divided by log 5 minus log of three is equal to 00:29:28.792 --> 00:29:35.266 X. OK, I've got X. You can put this lump of numbers here 00:29:35.266 --> 00:29:40.402 into a Calculator and work them out. One way of making that 00:29:40.402 --> 00:29:44.682 calculation alittle bit simpler though, is to notice that here 00:29:44.682 --> 00:29:49.390 we are subtracting two logs and if we're subtracting two logs, 00:29:49.390 --> 00:29:54.526 that means we are in fact dividing five by three in terms 00:29:54.526 --> 00:30:00.090 of the numbers themselves. So we can write this as two log 5. 00:30:00.140 --> 00:30:07.130 Over log of 5 / 3. 00:30:07.420 --> 00:30:11.290 And that gives us perhaps a slightly better form to 00:30:11.290 --> 00:30:12.838 do the calculations in. 00:30:14.230 --> 00:30:19.599 Now there's one further thing that I just want to have a look 00:30:19.599 --> 00:30:23.729 at and this will become quite important when doing calculus 00:30:23.729 --> 00:30:27.695 so. Let me take the number 00:30:27.695 --> 00:30:33.980 8. And I'm going to raise 2 to the power 8. 00:30:35.470 --> 00:30:38.402 Let me 00:30:38.402 --> 00:30:45.270 know. Take logs to base two 00:30:45.270 --> 00:30:48.966 of two to the 00:30:48.966 --> 00:30:55.400 power 8. Well, I know what that is, that's eight. 00:30:57.110 --> 00:31:00.608 I just think what's happened there. We started with a number. 00:31:01.450 --> 00:31:06.082 We took a base and we raise the base to that power. 00:31:07.280 --> 00:31:10.367 We then took logs to that bass. 00:31:11.080 --> 00:31:15.799 Of our resulting answer and we ended up with the number that we 00:31:15.799 --> 00:31:16.888 first started with. 00:31:18.160 --> 00:31:24.244 So any fact what we did here undid what we've done there. 00:31:25.150 --> 00:31:30.716 Now when that happens, what we're saying is we have got 00:31:30.716 --> 00:31:31.728 inverse operations. 00:31:32.750 --> 00:31:37.730 That what this operation does, raising the base to the 00:31:37.730 --> 00:31:42.212 given power is undone by what this operation does. 00:31:43.620 --> 00:31:47.526 Does it work the other way around? 00:31:49.090 --> 00:31:52.486 So let's start with eight again. 00:31:53.410 --> 00:31:57.110 This time, let's take the 00:31:57.110 --> 00:31:59.610 log. To base. 00:32:00.430 --> 00:32:02.938 Two of eight. 00:32:03.820 --> 00:32:07.940 Well, I know the log. 00:32:09.100 --> 00:32:13.552 To base two of eight, well, I need to write the 18 a different 00:32:13.552 --> 00:32:18.322 way I need it is 2 to the power three and then of course gives 00:32:18.322 --> 00:32:21.184 me the answer straight away. This number is 3. 00:32:22.590 --> 00:32:29.690 Now I take my base #2 and I raise it 00:32:29.690 --> 00:32:32.530 to this power 3. 00:32:33.400 --> 00:32:35.580 And the answers 8. 00:32:36.990 --> 00:32:41.973 So again, I've done these two operations in a different order. 00:32:42.790 --> 00:32:47.652 But the number that I started with and the number that I end 00:32:47.652 --> 00:32:49.896 up with are exactly the same. 00:32:50.860 --> 00:32:56.866 So what I've got here are two inverse operations. What one 00:32:56.866 --> 00:33:03.418 does the other one undoes? So in terms of our standard basis, 00:33:03.418 --> 00:33:10.516 what does that mean? It means that the natural log of E to 00:33:10.516 --> 00:33:17.080 the X. Here I've taken X as a power, and I've raised the 00:33:17.080 --> 00:33:22.240 base E to that power X and then I've taken the log. 00:33:22.770 --> 00:33:30.414 The log undoes what I did with the X and leaves me 00:33:30.414 --> 00:33:34.236 with X again. Similarly, if I 00:33:34.236 --> 00:33:42.060 Hav E. Raised to the power, the log of X, this is again X 00:33:42.060 --> 00:33:46.860 because I've used these two process is these two things. 00:33:46.860 --> 00:33:51.180 Again in combination there inverse operations, so they must 00:33:51.180 --> 00:33:54.060 undo what the other one has 00:33:54.060 --> 00:34:00.890 done. Similarly. If we take our base 10, 00:34:00.890 --> 00:34:02.510 then the log. 00:34:03.750 --> 00:34:10.966 Of 10 Raised to the power, X is just X. 00:34:11.780 --> 00:34:13.768 And if we take. 00:34:14.360 --> 00:34:21.238 10 And we raise it to the power. The log of X. Then 00:34:21.238 --> 00:34:25.578 again, the answer is just X because we have undone. 00:34:26.290 --> 00:34:32.335 By raising 10 to this power, we have undone what we did by 00:34:32.335 --> 00:34:36.055 taking the log of X to base 10. 00:34:37.940 --> 00:34:41.594 This is the particularly important one when you're doing 00:34:41.594 --> 00:34:46.872 calculus. This is the one that you will need to bear in mind.