A Few Words About Logarithms and Exponentials Help

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.

Fig. 1.58

Fig. 1.59

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.