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^x, when 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 = log₂ 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.