WEBVTT

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How does the difference between

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point 0000000398 and

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point 00000000398

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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

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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,

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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

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to the 3rd power, equals 8.

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The exponent p is said to be

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the logarithm of the number n.

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Most of the time this would be written:

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"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

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log base 10 of 10,000?

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The same question could be asked
using exponents:

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"10 raised to what power is 10,000?"

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Well, 10 to the 4th is 10,000.

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So, log base 10 of 10,000

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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

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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:

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the log button only computes
logarithms of base 10.

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What if you want to go into
computer science

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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

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and not others?

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Well, it leads us into an interesting

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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: 


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pH equals negative log base 10 of 
the hydrogen ion concentration, or H plus.

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We can find the pH of water samples

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with hydrogen ion concentration of 
point 0000000398

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and point 00000000398

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quickly on a calculator.

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Punch: negative log
of each of those numbers,

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and you'll see the pH's 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 .0000000398

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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 equals p"

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by repeating: "The base raised
to what power equals the number?"

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"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

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instead of eyedrops after swimming.