Logarithms, Explained - Steve Kelly
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0:15 - 0:18How does the difference between point
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0:18 - 0:210-0-0-0-0-0-0-3-9-8
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0:21 - 0:24and point 0-0-0-0-0-0-0-0-3-9-8
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0:24 - 0:28cause one to have red eyes after swimming?
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0:28 - 0:31To answer this, we first need a way of dealing with rather small numbers,
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0:31 - 0:34or in some cases extremely large numbers.
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0:34 - 0:37This leads us to the concept of logarithms.
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0:37 - 0:39Well, what are logarithms?
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0:39 - 0:41Let's take the base number - b - and raise it to a power, p,
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0:41 - 0:43like 2 to the 3rd power
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0:43 - 0:46and have it equal a number n.
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0:46 - 0:49We get an exponential equation b raised to the p power equals n.
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0:49 - 0:52In our example, that'd be 2 raised to the 3rd power equals 8.
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0:52 - 0:56The exponent p is said to be the logarithm of the number n.
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0:56 - 1:02Most of the time this would be written "log base b of a number equals p, the power."
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1:02 - 1:06This is starting to sound a bit confusing with all the variables,
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1:06 - 1:08so let's show this with an example.
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1:08 - 1:11What is the value of log base 10 of 10 thousand?
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1:11 - 1:14The same question could be asked using exponents.
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1:14 - 1:1710 raised to what power is 10 thousand?
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1:17 - 1:20Well, 10 to the 4th is 10 thousand. So, log base 10 of 10 thousand
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1:20 - 1:22must equal 4.
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1:22 - 1:26This example can also be completed very simply on a scientific calculator.
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1:26 - 1:29Log base 10 is used so frequently in the sciences
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1:29 - 1:34that it has the honor of having its own button on most calculators.
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1:34 - 1:37If the calculator will figure out logs for me,
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1:37 - 1:39why study them?
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1:39 - 1:43Just a quick reminder, the log button only computes logarithms of base 10.
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1:43 - 1:47What if you want to go into computer science and need to understand base 2?
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1:47 - 1:49So what is log base 2 of 64?
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1:49 - 1:53In other words, 2 raised to what power is 64?
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1:53 - 1:58Well, use your fingers. 2, 4, 8, 16, 32, 64.
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1:58 - 2:03So log base 2 of 64 must equal 6.
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2:03 - 2:05So what does this have to do with my eyes turning red
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2:05 - 2:08in some swimming pools and not others?
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2:08 - 2:12Well, it leads us into an interesting use of logarithms in chemistry:
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2:12 - 2:15finding the pH of water samples.
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2:15 - 2:17pH tells us how acidic or basic a sample is,
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2:17 - 2:22and can be calculated with the formula pH equals negative log base 10
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2:22 - 2:25of the hydrogen ion concentration, or H plus.
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2:25 - 2:29We can find the pH of water samples with hydrogen ion concentration of
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2:29 - 2:32point 0-0-0-0-0-0-0-3-9-8
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2:32 - 2:37and point 0-0-0-0-0-0-0-0-3-9-8
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2:37 - 2:40quickly on a calculator. Punch:
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2:40 - 2:45negative log of each of those numbers, and you'll see the pHs are 7.4 and 8.4.
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2:45 - 2:48Since the tears in our eyes have a pH of about 7.4,
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2:48 - 2:54the H plus concentration of .70398 will feel nice on your eyes.
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2:54 - 2:58But the pH of 8.4 will make you feel itchy and red.
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2:58 - 3:02It's easy to remember logarithms - log base b of some number n
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3:02 - 3:07equals p - by repeating "the base raised to what power equals the number?"
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3:07 - 3:12The base raised to what power equals the number? The base raised to what power equals the number?
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3:12 - 3:14So now we know logarithms are very powerful
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3:14 - 3:17when dealing with extremely small or large numbers.
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3:17 -Logarithms can even be used instead of eyedrops after swimming.
- Title:
- Logarithms, Explained - Steve Kelly
- Description:
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View full lesson: http://ed.ted.com/lessons/steve-kelly-logarithms-explained
What are logarithms and why are they useful? Get the basics on these critical mathematical functions -- and discover why smart use of logarithms can determine whether your eyes turn red at the swimming pool this summer.
Lesson by Steve Kelly, animation by TED-Ed.
- Video Language:
- English
- Team:
closed TED
- Project:
- TED-Ed
- Duration:
- 03:35
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Tulio Leao
Excellent transcription. I did, however, correct some parts of the subtitle to comply with the 21 char/s rule.
Best Regards
Retired user
Major mistake in the transcript at 02:49. The speaker says 'point seven Zeros 398" (= 0.0000000398), but the transcript says '0.70398'.
Tulio Leao
Thanks for finding this mistake Johanna. I've fixed it now :)