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Exponential Basics Help

By — McGraw-Hill Professional
Updated on Aug 31, 2011

Exponential Basics

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

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

Transcendental Functions 6.2 Exponential Basics

Fig. 6.4

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

Transcendental Functions 6.2 Exponential Basics

the function ln x is increasing. As a result,

Transcendental Functions 6.2 Exponential Basics

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.

Transcendental Functions 6.2 Exponential Basics

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 Transcendental Functions 6.2 Exponential Basics .

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

Transcendental Functions 6.2 Exponential Basics

Solution

We calculate that

Transcendental Functions 6.2 Exponential Basics

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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