0:00:15.000,0:00:18.000
How does the difference between point

0:00:18.000,0:00:21.000
0-0-0-0-0-0-0-3-9-8

0:00:21.000,0:00:24.000
and point 0-0-0-0-0-0-0-0-3-9-8

0:00:24.000,0:00:28.000
cause one to have red eyes after swimming?

0:00:28.000,0:00:31.000
To answer this, we first need a way of dealing with rather small numbers,

0:00:31.000,0:00:34.000
or in some cases extremely large numbers.

0:00:34.000,0:00:37.000
This leads us to the concept of logarithms.

0:00:37.000,0:00:39.000
Well, what are logarithms?

0:00:39.000,0:00:41.000
Let's take the base number - b - and raise it to a power, p,

0:00:41.000,0:00:43.000
like 2 to the 3rd power

0:00:43.000,0:00:46.000
and have it equal a number n.

0:00:46.000,0:00:49.000
We get an exponential equation b raised to the p power equals n.

0:00:49.000,0:00:52.000
In our example, that'd be 2 raised to the 3rd power equals 8.

0:00:52.000,0:00:56.000
The exponent p is said to be the logarithm of the number n.

0:00:56.000,0:01:02.000
Most of the time this would be written "log base b of a number equals p, the power."

0:01:02.000,0:01:06.000
This is starting to sound a bit confusing with all the variables,

0:01:06.000,0:01:08.000
so let's show this with an example.

0:01:08.000,0:01:11.000
What is the value of log base 10 of 10 thousand?

0:01:11.000,0:01:14.000
The same question could be asked using exponents.

0:01:14.000,0:01:17.000
10 raised to what power is 10 thousand?

0:01:17.000,0:01:20.000
Well, 10 to the 4th is 10 thousand. So, log base 10 of 10 thousand

0:01:20.000,0:01:22.000
must equal 4.

0:01:22.000,0:01:26.000
This example can also be completed very simply on a scientific calculator.

0:01:26.000,0:01:29.000
Log base 10 is used so frequently in the sciences

0:01:29.000,0:01:34.000
that it has the honor of having its own button on most calculators.

0:01:34.000,0:01:37.000
If the calculator will figure out logs for me,

0:01:37.000,0:01:39.000
why study them?

0:01:39.000,0:01:43.000
Just a quick reminder, the log button only computes logarithms of base 10.

0:01:43.000,0:01:47.000
What if you want to go into computer science and need to understand base 2?

0:01:47.000,0:01:49.000
So what is log base 2 of 64?

0:01:49.000,0:01:53.000
In other words, 2 raised to what power is 64?

0:01:53.000,0:01:58.000
Well, use your fingers. 2, 4, 8, 16, 32, 64.

0:01:58.000,0:02:03.000
So log base 2 of 64 must equal 6.

0:02:03.000,0:02:05.000
So what does this have to do with my eyes turning red

0:02:05.000,0:02:08.000
in some swimming pools and not others?

0:02:08.000,0:02:12.000
Well, it leads us into an interesting use of logarithms in chemistry:

0:02:12.000,0:02:15.000
finding the pH of water samples.

0:02:15.000,0:02:17.000
pH tells us how acidic or basic a sample is,

0:02:17.000,0:02:22.000
and can be calculated with the formula pH equals negative log base 10

0:02:22.000,0:02:25.000
of the hydrogen ion concentration, or H plus.

0:02:25.000,0:02:29.000
We can find the pH of water samples with hydrogen ion concentration of

0:02:29.000,0:02:32.000
point 0-0-0-0-0-0-0-3-9-8

0:02:32.000,0:02:37.000
and point 0-0-0-0-0-0-0-0-3-9-8

0:02:37.000,0:02:40.000
quickly on a calculator. Punch:

0:02:40.000,0:02:45.000
negative log of each of those numbers, and you'll see the pHs are 7.4 and 8.4.

0:02:45.000,0:02:48.000
Since the tears in our eyes have a pH of about 7.4,

0:02:48.000,0:02:54.000
the H plus concentration of .70398 will feel nice on your eyes.

0:02:54.000,0:02:58.000
But the pH of 8.4 will make you feel itchy and red.

0:02:58.000,0:03:02.000
It's easy to remember logarithms - log base b of some number n

0:03:02.000,0:03:07.000
equals p - by repeating "the base raised to what power equals the number?"

0:03:07.000,0:03:12.000
The base raised to what power equals the number? The base raised to what power equals the number?

0:03:12.000,0:03:14.000
So now we know logarithms are very powerful

0:03:14.000,0:03:17.000
when dealing with extremely small or large numbers.

0:03:17.000,9:59:59.000
Logarithms can even be used instead of eyedrops after swimming.