Logarithms, Explained - Steve Kelly

How does the difference between point 0000000398 and point 00000000398 cause one to have red eyes after swimming?

To answer this, we first need a way of dealing with rather small numbers, or in some cases extremely large numbers. This leads us to the concept of logarithms.

Well, what are logarithms? Let's take the base number, b, and raise it to a power, p, like 2 to the 3rd power, and have it equal a number n. We get an exponential equation: b raised to the p power equals n. In our example, that'd be 2 raised to the 3rd power equals 8.

The exponent p is said to be the logarithm of the number n. Most of the time this would be written: "log, base b, of a number equals p, the power." This is starting to sound a bit confusing with all the variables, so let's show this with an example.

What is the value of log base 10 of 10,000? The same question could be asked using exponents: "10 raised to what power is 10,000?" Well, 10 to the 4th is 10,000. So, log base 10 of 10,000 must equal 4.

This example can also be completed very simply on a scientific calculator. Log base 10 is used so frequently in the sciences that it has the honor of having its own button on most calculators. If the calculator will figure out logs for me, why study them?

Just a quick reminder: the log button only computes logarithms of base 10. What if you want to go into computer science and need to understand base 2? So what is log base 2 of 64? In other words, 2 raised to what power is 64?

Well, use your fingers. 2, 4, 8, 16, 32, 64. So log base 2 of 64 must equal 6.

So what does this have to do with my eyes turning red in some swimming pools and not others? Well, it leads us into an interesting use of logarithms in chemistry: finding the pH of water samples.

pH tells us how acidic or basic a sample is, and can be calculated with the formula: pH equals negative log base 10 of the hydrogen ion concentration, or H plus. We can find the pH of water samples with hydrogen ion concentration of point 0000000398 and point 00000000398 quickly on a calculator.

Punch: negative log of each of those numbers, and you'll see the pH's are 7.4 and 8.4. Since the tears in our eyes have a pH of about 7.4, the H plus concentration of .0000000398 will feel nice on your eyes, but the pH of 8.4 will make you feel itchy and red.

It's easy to remember logarithms "log base b of some number n equals p" by repeating: "The base raised to what power equals the number?" "The BASE raised to what POWER equals the NUMBER?"

So now we know logarithms are very powerful when dealing with extremely small or large numbers. Logarithms can even be used instead of eyedrops after swimming.