## Calculus AB and Calculus BC

## CHAPTER 1 Functions

### E. EXPONENTIAL AND LOGARITHMIC FUNCTIONS

**E1. Exponential Functions.**

The following laws of exponents hold for all rational *m* and *n*, provided that *a* > 0, *a* ≠ 1:

The exponential function *f* (*x*) = *a** ^{x}* (

*a*> 0,

*a*≠ 1) is thus defined for all real

*x*; its domain is the set of positive reals. The graph of

*y*=

*a*

*, when*

^{x}*a*= 2, is shown in Figure N1–8.

Of special interest and importance in the calculus is the exponential function *f* (*x*) = *e** ^{x}*, where

*e*is an irrational number whose decimal approximation to five decimal places is 2.71828.

**E2. Logarithmic Functions.**

Since *f* (*x*) = *a** ^{x}* is one-to-one, it has an inverse,

*f*

^{−1}(

*x*) = log

_{a}*x*, called the

*logarithmic*function with base

*a*. We note that

*y* = log_{a}*x* if and only if *a** ^{y}* =

*x*.

The domain of log _{a}*x* is the set of positive reals; its range is the set of all reals. It follows that the graphs of the pair of mutually inverse functions *y* = 2* ^{x}* and

*y*= log

_{2}

*x*are symmetric to the line

*y*=

*x*, as can be seen in Figure N1–8.

**FIGURE N1–8**

The logarithmic function log _{a}*x* (*a* > 0, *a* ≠ 1) has the following properties:

The logarithmic base *e* is so important and convenient in calculus that we use a special symbol:

log _{e}*x* = ln *x*.

Logarithms with base *e* are called *natural* logarithms. The domain of ln *x* is the set of positive reals; its range is the set of all reals. The graphs of the mutually inverse functions ln *x* and *e** ^{x}* are given in the Appendix.