Page 168 - Calculus Demystified

CHAPTER 6
Transcendental Functions
The inverse function to the natural logarithm function is called the exponential 155
function and is written exp(x). The domain of exp is the entire real line. The range
is the set of positive real numbers.
EXAMPLE 6.7
Using the definition of the exponential function, simplify the expressions
exp(ln a + ln b) and ln(7 ·[exp(c)]).

SOLUTION
We use the key property that the exponential function is the inverse of the
logarithm function. We have

exp(ln a + ln b) = exp(ln(a · b)) = a · b,
ln(7 ·[exp(c)]) = ln 7 + ln(exp(c)) = ln 7 + c.

3
5
You Try It: Simplify the expression ln(a · 3 · 5 −4 ).
6.2.1 FACTS ABOUT THE EXPONENTIAL FUNCTION

First review the properties of inverse functions that we learned in Subsection 1.8.5.
The graph of exp(x) is obtained by reflecting the graph of ln x in the line y = x.
We exhibit the graph of y = exp(x) in Fig. 6.5.

Fig. 6.5

We see, from inspection of this figure, that exp(x) is increasing and is concave
up. Since ln(1) = 0 we may conclude that exp(0) = 1. Next we turn to some of
the algebraic properties of the exponential function.