0:00:15.550,0:00:17.030
How does the difference between

0:00:17.030,0:00:20.860
point 0000000398 and

0:00:20.860,0:00:25.660
point 00000000398

0:00:25.660,0:00:27.880
cause one to have red eyes after swimming?

0:00:27.880,0:00:29.980
To answer this, we first need a way of

0:00:29.980,0:00:32.000
dealing with rather small numbers,

0:00:32.000,0:00:34.000
or in some cases extremely large numbers.

0:00:34.000,0:00:36.310
This leads us[br]to the concept of logarithms.

0:00:36.310,0:00:37.980
Well, what are logarithms?

0:00:37.980,0:00:40.033
Let's take the base number, b,

0:00:40.033,0:00:41.536
and raise it to a power, p,

0:00:41.536,0:00:43.000
like 2 to the 3rd power,

0:00:43.000,0:00:45.570
and have it equal a number n.

0:00:45.570,0:00:49.270
We get an exponential equation:[br]b raised to the p power, equals n.

0:00:49.270,0:00:51.366
In our example, that'd be 2 raised

0:00:51.366,0:00:53.192
to the 3rd power, equals 8.

0:00:53.192,0:00:55.041
The exponent p is said to be

0:00:55.041,0:00:57.200
the logarithm of the number n.

0:00:57.200,0:00:59.140
Most of the time this would be written:

0:00:59.140,0:01:03.540
"log, base b, of a number[br]equals p, the power."

0:01:03.540,0:01:06.593
This is starting to sound a bit confusing[br]with all the variables,

0:01:06.593,0:01:08.220
so let's show this with an example.

0:01:08.220,0:01:09.333
What is the value of

0:01:09.333,0:01:11.676
log base 10 of 10,000?

0:01:11.676,0:01:14.000
The same question could be asked[br]using exponents:

0:01:14.000,0:01:16.380
"10 raised to what power is 10,000?"

0:01:16.380,0:01:18.666
Well, 10 to the 4th is 10,000.

0:01:18.666,0:01:20.332
So, log base 10 of 10,000

0:01:20.332,0:01:22.290
must equal 4.

0:01:22.290,0:01:26.310
This example can also be completed [br]very simply on a scientific calculator.

0:01:26.310,0:01:28.446
Log base 10 is used so frequently

0:01:28.446,0:01:29.712
in the sciences

0:01:29.712,0:01:34.790
that it has the honor of having[br]its own button on most calculators.

0:01:34.790,0:01:37.000
If the calculator will figure out[br]logs for me,

0:01:37.000,0:01:38.470
why study them?

0:01:38.470,0:01:39.756
Just a quick reminder:

0:01:39.756,0:01:43.532
the log button only computes[br]logarithms of base 10.

0:01:43.532,0:01:45.604
What if you want to go into[br]computer science

0:01:45.604,0:01:47.746
and need to understand base 2?

0:01:47.746,0:01:50.190
So what is log base 2 of 64?

0:01:50.190,0:01:53.990
In other words,[br]2 raised to what power is 64?

0:01:53.990,0:01:59.110
Well, use your fingers.[br]2, 4, 8, 16, 32, 64.

0:01:59.110,0:02:03.510
So log base 2 of 64 must equal 6.

0:02:03.510,0:02:06.370
So what does this have to do[br]with my eyes turning red

0:02:06.370,0:02:07.696
in some swimming pools

0:02:07.696,0:02:08.812
and not others?

0:02:08.812,0:02:10.601
Well, it leads us into an interesting

0:02:10.601,0:02:12.530
use of logarithms in chemistry:

0:02:12.530,0:02:14.600
finding the pH of water samples.

0:02:14.600,0:02:17.600
pH tells us how acidic[br]or basic a sample is,

0:02:17.600,0:02:19.600
and can be calculated with the formula: [br]

0:02:19.600,0:02:25.630
pH equals negative log base 10 of [br]the hydrogen ion concentration, or H plus.

0:02:25.630,0:02:27.980
We can find the pH of water samples

0:02:27.980,0:02:33.210
with hydrogen ion concentration of [br]point 0000000398

0:02:33.210,0:02:38.620
and point 00000000398

0:02:38.620,0:02:39.870
quickly on a calculator.

0:02:39.870,0:02:41.986
Punch: negative log[br]of each of those numbers,

0:02:41.986,0:02:46.282
and you'll see the pH's are 7.4 and 8.4.

0:02:46.282,0:02:49.150
Since the tears in our eyes[br]have a pH of about 7.4,

0:02:49.150,0:02:53.420
the H plus concentration of .70398

0:02:53.420,0:02:54.960
will feel nice on your eyes,

0:02:54.960,0:02:59.260
but the pH of 8.4[br]will make you feel itchy and red.

0:02:59.260,0:03:04.030
It's easy to remember logarithms[br]"log base b of some number n equals p"

0:03:04.030,0:03:07.733
by repeating: "The base raised[br]to what power equals the number?"

0:03:07.733,0:03:12.506
"The BASE raised to what POWER[br]equals the NUMBER?"

0:03:12.506,0:03:14.890
So now we know[br]logarithms are very powerful

0:03:14.890,0:03:18.056
when dealing with[br]extremely small or large numbers.

0:03:18.056,0:03:19.692
Logarithms can even be used

0:03:19.692,0:03:21.928
instead of eyedrops after swimming.