We continue this topic after the intermediate value theorem posted earlier.
For , define
by
. It is easily seen that
if
. This shows that f is one to one. Further,
, whereas
. That,
is onto
follows from the intermediate value theorem. Thus,
defined by
is bijective. So there is a unique map
such that for every y in
and
for every x in
.
This function g is what we call the logarithm function of y to the base a, written as . In fact, the logarithm is a continuous function.
For ,
, let
, where
. Then, we have for
,
, or
or
Exercise.
If is an increasing continuous function, show that it is bijective onto its range and its inverse is also continuous.
With the help of the logarithm function, we can evaluate .
Let so that
as
. Also,
. So, we have
, that is,
.
In the step before last, we have used the fact that the logarithm is a continuous function and that , while in the last step we have observed that
(Exercise).
More later,
Nalin Pithwa