**Exponential Basics**

Examine Fig. 6.4, which shows the graph of the function

*f*(*x*) = ln *x*, *x* > 0.

As we observed in Section 1, the function *f* takes on all real values. We already have noticed that, since

the function ln *x* is increasing. As a result,

is one-to-one and onto. Hence the natural logarithm function has an inverse.

The inverse function to the natural logarithm function is called the *exponential 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**

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* .

**You Try It** : Simplify the expression ln ( *a* ^{3} · 3 ^{5} · 5 ^{−4} ).

**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.

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.

For all real numbers *a* and *b* we have

(a) exp( *a* + *b* ) = [exp( *a* )] · [exp ( *b* )].

(b) For any *a* and *b* we have .

These properties are verified just by exploiting the fact that the exponential is the inverse of the logarithm, as we saw in Example 6.7.

**Example**

Use the basic properties to simplify the expression

**Solution**

We calculate that

**You Try It** : Simplify the expression (exp *a* ) ^{−3} · (exp *b* ) ^{2} / exp(5 *c* ).

Find practice problems and solutions for these concepts at: Transcendental Functions Practice Test.

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