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Index
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:
12.43
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