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12.5 Logarithms

12.37   Definition (Logarithm.) Let . The logarithm of is the unique number such that . We denote the logarithm of by , Hence
 (12.38)

12.39   Remark. Since is the unique number such that , it follows that
 (12.40)

12.41   Theorem. For all ,

Proof:

12.42   Exercise. A Show that
a) for all .
b) for all , .
c) for all .

12.43   Remark. It follows from the fact that is strictly increasing on that is strictly increasing on : if , then both of the statements and lead to contradictions.

12.44   Theorem (Continuity of .) is a continuous function on .

Proof: Let , and let be a sequence in such that . I want to show that . Let be a precision function for . I want to construct a precision function for .

Scratchwork: For all , and all ,

Note that since is strictly increasing, and are both positive. This calculation motivates the following definition:

For all , let

Then for all , ,

Hence is a precision function for .

12.45   Theorem (Differentiability of .)The function is differentiable on and

Proof: Let and let be a sequence in . Then

(Note, I have not divided by .) Since is continuous, I know , and hence

Hence,

i.e.,

This shows that .

Next: 12.6 Trigonometric Functions Up: 12. Power Series Previous: 12.4 The Exponential Function   Index