Gregory Schmidt                                                                                                    

Write up #1                                                                                                      

Composition of the

Logarithmic and Exponential




Problem: Let  and










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 .







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.









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 .







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







Well, as expected  and .


What can we guess about  and , where .


This time  and .





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 .