Exponential Basics Help
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.
Using the definition of the exponential function, simplify the expressions
exp(ln a + ln b ) and ln(7 · [exp( c )]).
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.
Use the basic properties to simplify the expression
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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