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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