**Introduction to a A Few Words About Logarithms and Exponentials**

We will give a more thorough treatment of the logarithm and exponential functions in Chapter 6. For the moment we record a few simple facts so that we may use these functions in the sections that immediately follow.

The logarithm is a function that is characterized by the property that

log( *x* · *y* ) = log *x* + log *y* .

It follows from this property that

log( *x* / *y* ) = log *x* − log *y*

and

log( *x* ^{n} ) = *n* · log *x*.

It is useful to think of log _{a} *b* as the power to which we raise *a* to get *b* , for any *a* , *b* > 0. For example, log _{2} 8 = 3 and log _{3} (1/27) = −3. This introduces the idea of the *logarithm to a base*.

**You Try It:** Calculate log _{5} 125, log _{3} (1/81), log _{2} 16.

The most important base for the logarithm is Euler’s number *e* ≈ 2.71828 .... Then we write ln *x* = log _{e} *x* . For the moment we take the logarithm to the base *e* , or the *natural logarithm* , to be given. It is characterized among all logarithm functions by the fact that its graph has tangent line with slope 1 at *x* = 1. See Fig. 1.58. Then we set

Note that this formula gives immediately that log _{e} *x* = In *x* , once we accept that log _{e} *e* = 1.

**Math Note:** In mathematics, we commonly write log x to mean the natural logarithm. Thus you will sometimes encounter In x and sometimes encounter log x (without any subscript); they are both understood to mean log _{e} x , the natural logarithm.

The exponential function exp *x* is defined to be the inverse function to In *x*. Figure 1.59 shows the graph of *y* = exp *x* . In fact we will see later that exp *x* = *e* ^{x}. More generally, the function *a* ^{x} is the inverse function to log _{a} *x*. The exponential has these properties:

(a) *a* ^{b + c} = *a* ^{b} · *a* ^{c};

These are really just restatements of properties of the logarithm function that we have already considered.

**You Try It:** Simplify the expressions 3 ^{2} · 5 ^{4} /(15) ^{3} and 2 ^{4} · 6 ^{3} · 12 ^{−4}.

Find practice problems and solutions for these concepts at: Calculus Basics Practice Test.

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