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Logarithms, Explained - Steve Kelly

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    How does the difference between point
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    0-0-0-0-0-0-0-3-9-8
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    and point 0-0-0-0-0-0-0-0-3-9-8
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    cause one to have red eyes after swimming?
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    To answer this, we first need a way of dealing with rather small numbers,
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    or in some cases extremely large numbers.
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    This leads us to the concept of logarithms.
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    Well, what are logarithms?
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    Let's take the base number - b - and raise it to a power, p,
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    like 2 to the 3rd power
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    and have it equal a number n.
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    We get an exponential equation b raised to the p power equals n.
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    In our example, that'd be 2 raised to the 3rd power equals 8.
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    The exponent p is said to be the logarithm of the number n.
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    Most of the time this would be written "log base b of a number equals p, the power."
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    This is starting to sound a bit confusing with all the variables,
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    so let's show this with an example.
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    What is the value of log base 10 of 10 thousand?
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    The same question could be asked using exponents.
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    10 raised to what power is 10 thousand?
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    Well, 10 to the 4th is 10 thousand. So, log base 10 of 10 thousand
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    must equal 4.
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    This example can also be completed very simply on a scientific calculator.
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    Log base 10 is used so frequently in the sciences
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    that it has the honor of having its own button on most calculators.
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    If the calculator will figure out logs for me,
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    why study them?
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    Just a quick reminder, the log button only computes logarithms of base 10.
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    What if you want to go into computer science and need to understand base 2?
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    So what is log base 2 of 64?
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    In other words, 2 raised to what power is 64?
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    Well, use your fingers. 2, 4, 8, 16, 32, 64.
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    So log base 2 of 64 must equal 6.
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    So what does this have to do with my eyes turning red
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    in some swimming pools and not others?
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    Well, it leads us into an interesting use of logarithms in chemistry:
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    finding the pH of water samples.
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    pH tells us how acidic or basic a sample is,
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    and can be calculated with the formula pH equals negative log base 10
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    of the hydrogen ion concentration, or H plus.
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    We can find the pH of water samples with hydrogen ion concentration of
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    point 0-0-0-0-0-0-0-3-9-8
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    and point 0-0-0-0-0-0-0-0-3-9-8
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    quickly on a calculator. Punch:
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    negative log of each of those numbers, and you'll see the pHs are 7.4 and 8.4.
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    Since the tears in our eyes have a pH of about 7.4,
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    the H plus concentration of .70398 will feel nice on your eyes.
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    But the pH of 8.4 will make you feel itchy and red.
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    It's easy to remember logarithms - log base b of some number n
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    equals p - by repeating "the base raised to what power equals the number?"
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    The base raised to what power equals the number? The base raised to what power equals the number?
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    So now we know logarithms are very powerful
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    when dealing with extremely small or large numbers.
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    Logarithms can even be used instead of eyedrops after swimming.
Title:
Logarithms, Explained - Steve Kelly
Description:

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.

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Video Language:
English
Team:
closed TED
Project:
TED-Ed
Duration:
03:35
  • Excellent transcription. I did, however, correct some parts of the subtitle to comply with the 21 char/s rule.

    Best Regards

  • Major mistake in the transcript at 02:49. The speaker says 'point seven Zeros 398" (= 0.0000000398), but the transcript says '0.70398'.

  • Thanks for finding this mistake Johanna. I've fixed it now :)

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