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14.29
Example.
We will now derive some properties of the
inverse function
![$E$](img165.gif)
of the logarithm.
We have
For all
and
in R,
If we apply
![$E$](img165.gif)
to both sides of this equality we get
For all
![$a\in\mbox{{\bf R}}$](img1436.gif)
we have
from which it follows that
If
![$a\in\mbox{{\bf R}}$](img1436.gif)
and
![$q \in \mbox{{\bf Q}}$](img3572.gif)
we have
If we apply
![$E$](img165.gif)
to both sides of this identity we get
In particular,
![\begin{displaymath}e^q = (E(1))^q = E(q) \mbox{ for all }q\in\mbox{{\bf Q}}.
\end{displaymath}](img3575.gif) |
(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
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
![$\displaystyle {e^{a-b} = {e^a \over e^b} \mbox{ for all }a,b \in \mbox{{\bf R}}}$](img3597.gif)
.
14.35
Exercise.
Show that if
![$a \in \mbox{${\mbox{{\bf R}}}^{+}$}$](img309.gif)
and
![$q \in \mbox{{\bf Q}}$](img3572.gif)
, 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