1 00:00:15,000 --> 00:00:18,000 How does the difference between point 2 00:00:18,000 --> 00:00:21,000 0-0-0-0-0-0-0-3-9-8 3 00:00:21,000 --> 00:00:24,000 and point 0-0-0-0-0-0-0-0-3-9-8 4 00:00:24,000 --> 00:00:28,000 cause one to have red eyes after swimming? 5 00:00:28,000 --> 00:00:31,000 To answer this, we first need a way of dealing with rather small numbers, 6 00:00:31,000 --> 00:00:34,000 or in some cases extremely large numbers. 7 00:00:34,000 --> 00:00:37,000 This leads us to the concept of logarithms. 8 00:00:37,000 --> 00:00:39,000 Well, what are logarithms? 9 00:00:39,000 --> 00:00:41,000 Let's take the base number - b - and raise it to a power, p, 10 00:00:41,000 --> 00:00:43,000 like 2 to the 3rd power 11 00:00:43,000 --> 00:00:46,000 and have it equal a number n. 12 00:00:46,000 --> 00:00:49,000 We get an exponential equation b raised to the p power equals n. 13 00:00:49,000 --> 00:00:52,000 In our example, that'd be 2 raised to the 3rd power equals 8. 14 00:00:52,000 --> 00:00:56,000 The exponent p is said to be the logarithm of the number n. 15 00:00:56,000 --> 00:01:02,000 Most of the time this would be written "log base b of a number equals p, the power." 16 00:01:02,000 --> 00:01:06,000 This is starting to sound a bit confusing with all the variables, 17 00:01:06,000 --> 00:01:08,000 so let's show this with an example. 18 00:01:08,000 --> 00:01:11,000 What is the value of log base 10 of 10 thousand? 19 00:01:11,000 --> 00:01:14,000 The same question could be asked using exponents. 20 00:01:14,000 --> 00:01:17,000 10 raised to what power is 10 thousand? 21 00:01:17,000 --> 00:01:20,000 Well, 10 to the 4th is 10 thousand. So, log base 10 of 10 thousand 22 00:01:20,000 --> 00:01:22,000 must equal 4. 23 00:01:22,000 --> 00:01:26,000 This example can also be completed very simply on a scientific calculator. 24 00:01:26,000 --> 00:01:29,000 Log base 10 is used so frequently in the sciences 25 00:01:29,000 --> 00:01:34,000 that it has the honor of having its own button on most calculators. 26 00:01:34,000 --> 00:01:37,000 If the calculator will figure out logs for me, 27 00:01:37,000 --> 00:01:39,000 why study them? 28 00:01:39,000 --> 00:01:43,000 Just a quick reminder, the log button only computes logarithms of base 10. 29 00:01:43,000 --> 00:01:47,000 What if you want to go into computer science and need to understand base 2? 30 00:01:47,000 --> 00:01:49,000 So what is log base 2 of 64? 31 00:01:49,000 --> 00:01:53,000 In other words, 2 raised to what power is 64? 32 00:01:53,000 --> 00:01:58,000 Well, use your fingers. 2, 4, 8, 16, 32, 64. 33 00:01:58,000 --> 00:02:03,000 So log base 2 of 64 must equal 6. 34 00:02:03,000 --> 00:02:05,000 So what does this have to do with my eyes turning red 35 00:02:05,000 --> 00:02:08,000 in some swimming pools and not others? 36 00:02:08,000 --> 00:02:12,000 Well, it leads us into an interesting use of logarithms in chemistry: 37 00:02:12,000 --> 00:02:15,000 finding the pH of water samples. 38 00:02:15,000 --> 00:02:17,000 pH tells us how acidic or basic a sample is, 39 00:02:17,000 --> 00:02:22,000 and can be calculated with the formula pH equals negative log base 10 40 00:02:22,000 --> 00:02:25,000 of the hydrogen ion concentration, or H plus. 41 00:02:25,000 --> 00:02:29,000 We can find the pH of water samples with hydrogen ion concentration of 42 00:02:29,000 --> 00:02:32,000 point 0-0-0-0-0-0-0-3-9-8 43 00:02:32,000 --> 00:02:37,000 and point 0-0-0-0-0-0-0-0-3-9-8 44 00:02:37,000 --> 00:02:40,000 quickly on a calculator. Punch: 45 00:02:40,000 --> 00:02:45,000 negative log of each of those numbers, and you'll see the pHs are 7.4 and 8.4. 46 00:02:45,000 --> 00:02:48,000 Since the tears in our eyes have a pH of about 7.4, 47 00:02:48,000 --> 00:02:54,000 the H plus concentration of .70398 will feel nice on your eyes. 48 00:02:54,000 --> 00:02:58,000 But the pH of 8.4 will make you feel itchy and red. 49 00:02:58,000 --> 00:03:02,000 It's easy to remember logarithms - log base b of some number n 50 00:03:02,000 --> 00:03:07,000 equals p - by repeating "the base raised to what power equals the number?" 51 00:03:07,000 --> 00:03:12,000 The base raised to what power equals the number? The base raised to what power equals the number? 52 00:03:12,000 --> 00:03:14,000 So now we know logarithms are very powerful 53 00:03:14,000 --> 00:03:17,000 when dealing with extremely small or large numbers. 54 00:03:17,000 --> 99:59:59,999 Logarithms can even be used instead of eyedrops after swimming.