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3.5.1 What Are Logarithms?

Essentially, a logarithm is just part of another (different) way of writing a number. You're used (probably) to writing numbers as fractions, for instance 1/3, and as decimal numbers, for instance

0.3333333333... Logarithms employ a third and less common way, which consists of writing the

number as one number {the base) raised to the power of another number (the exponent). Instead of writing 8, for instance, we could write 2^3 , which means “2 multiplied by itself three times”, or 2 ×2 ×2. In this case, 2 is the base and 3 is the exponent. We could also write 25 as 5^2, or 5 ×5. 5 is the base here, and 2 is the exponent. Usually, when we talk about logs, we talk in terms of exponents with a specific base. There are two bases that are commonly used in chemistry: “e”, and 10. “e” is a nonrepeating decimal number, the first 11 or so digits of which are 2.718281828459. 10, of course, is 10. We can write numbers as a combination of either one of these two bases raised to some power (which

will change with the choice of base and the number we wish to represent, of course). How does all the above relate to logs? Well, when you take the log of a number, you are finding the exponent to which either “e” or 10 must be raised in order to recreate the number of which you took the log. You must be specific about the kind of log you are taking too. When you want to use “e” as a base, you say that you are taking the natural logarithmof the number. When you want to use 10 as a base, you say that you are taking the logarithm base 10of that number. These are accessed on your calculator via different buttons, too. Usually, to take the natural logarithm of a number, you use the button marked lnxon your calculator. The button marked logxis for when you want to use base 10 for your logarithms.

This, by now, is probably monstrously confusing, so let's take a look at a few examples to clear things up. Suppose you wanted to take the natural log of 9 (for whatever reason). If you punched it up on the calculator, you would hit the sequence:

9 [lnx].

You would then see the answer (which is 2.197225...). That means that e^2.197225...= 9. If you were to use base 10, you would use the keystrokes:

9 [logx]

and see the answer (which is 0.9542425...). That means that 10^0.9542425...= 9.

Earlier on, in the introduction to this section, I used whole number bases and exponents. Don't worry about how you would multiply anything by itself a fractional number of times; your calculator will deal with that for you.