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12.37
Definition (Logarithm.)
Let
![$t\in\mbox{${\mbox{{\bf R}}}^{+}$}$](img1849.gif)
. The
logarithm of
is the unique number
![$s\in\mbox{{\bf R}}$](img1851.gif)
such that
![$e^s=t$](img1864.gif)
. We denote the logarithm of
![$t$](img920.gif)
by
![$\ln(t)$](img1865.gif)
, Hence
![\begin{displaymath}
e^{\ln (t)}=t\mbox{ for all }t\in\mbox{${\mbox{{\bf R}}}^{+}$}.
\end{displaymath}](img1866.gif) |
(12.38) |
12.39
Remark.
Since
![$\ln (e^r)$](img1867.gif)
is the unique number
![$s$](img408.gif)
such that
![$e^s=e^r$](img1868.gif)
, it follows that
![\begin{displaymath}
\ln(e^r)=r\mbox{ for all }r\in\mbox{{\bf R}}.
\end{displaymath}](img1869.gif) |
(12.40) |
12.41
Theorem.
For all
,
Proof:
12.43
Remark.
It follows from the fact that
![$\exp$](img1831.gif)
is strictly increasing on
![$\mbox{{\bf R}}$](img4.gif)
that
![$\ln$](img1877.gif)
is
strictly increasing on
![$\mbox{${\mbox{{\bf R}}}^{+}$}$](img75.gif)
: if
![$0<t<s$](img1878.gif)
, then both of the statements
![$\ln(t)=\ln(s)$](img1879.gif)
and
![$\ln(t)>\ln(s)$](img1880.gif)
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
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