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# 14.4 The Exponential Function

14.29   Example. We will now derive some properties of the inverse function of the logarithm.

We have

For all and in R,

If we apply to both sides of this equality we get

For all we have

from which it follows that

If and we have

If we apply to both sides of this identity we get

In particular,
 (14.30)

Now we have defined for all , but we have only defined when and . (We know what is, but we have not defined .) Because of relation (14.30) we often write in place of . is called the exponential function, and is written

We can summarize the results of this example in the following theorem:

14.31   Theorem (Properties of the exponential function.) The exponential function is a function from R onto . We have
 (14.32) (14.33)

Proof: We have proved all of these properties except for relation (14.32). The proof of (14.32) is the next exercise.

14.34   Exercise. Show that .

14.35   Exercise. Show that if and , then

14.36   Definition (.) The result of the last exercise motivates us to make the definition

14.37   Exercise. Prove the following results:

Next: 14.5 Inverse Function Theorems Up: 14. The Inverse Function Previous: 14.3 Inverse Functions   Index
Ray Mayer 2007-09-07