Algebra Know-It-ALL

Logarithms and exponentials show up in many branches of mathematics, sometimes unex-

pectedly. If you’re getting ready for more advanced subjects such as analysis, calculus, or engi-

neering mathematics, you should know something about logarithms and exponentials.

What Is a Logarithm?

Alogarithm (sometimes called a log) of a quantity is a power to which a positive real constant

is raised to get that quantity. The constant is called the base, which is usually 10 (the base

familiar to most of us) or e, an irrational number called Euler’s constant.

Findinge

If your calculator lacks a key that displays e directly, here’s a trick that will put it there for you.

Enter the number 1, then hit the inverse-function key or put a check in the appropriate box if

you’re using a computer (it might be labeled “Inv” or “Rev”). Then hit the “ln” or “loge” key.

You should see 2.71828 ... and then some more digits.

The log base

Imagine a positive real constant b. We raise b to some real variable power y, getting another

real number x as the result. We can write this as

by=x

where b > 0. In this equation, the exponent y is the base-b logarithm of x. When we say it that

way, we write it as

y= logbx

A logarithm is an exponent in a situation where it is also the dependent variable in a function.

The above equation tells us that y is a base-b logarithmic function of x.

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CHAPTER

29 Logarithms and Exponentials

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