1

Gregory Schmidt

Write up #1

Composition of the

Logarithmic and Exponential

Functions

Problem: Let and

Consider:

(i)

(ii)

(iii)

(iv)

We will explore the graphs of our new functions, and explore the changes in the domains and ranges.

First we consider the graphs of and .

(1)

We see immediately, that and .

We notice that appears to remain to the right of the y-axis, and appears to remain above the x-axis.

Just a little thought reveals that in fact this must be the case since for all , This is due to the fact that , and so any power of must also be greater than zero.

Now, for , we must first note how and are related. When we talk about , we are really just asking what power of is equal to . That is, when

, what is ?

For example, since implies that This explains why is always to the right of the y-axis, since for all . Hence, is not defined for .

We say that the domain of , denoted and the range of , denoted

Similarly, and .

Now, , like , is always to the right of the y-axis.

(2)

Why?

Well, obviously if , since is not defined for .

Hence, and .

We also not that lies between the graphs of and , but this makes sense because we are simply adding the two respective functions to form .

Now consider .

(3)

Again we see that and .

But this time the growth of eventually overtakes . Just a little thought we understand why this must be the case, since for all , and if .

What about

(4)

Well, as expected and .

What can we guess about and , where .

This time and .

(5)

Notice: appears to be very similar to the graph . Well, we need only note that , and so , the identity function.

In this case, we say that and , and so .