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