
Exponents and logarithms: inverse functions
Let's begin with a concrete question.
Suppose that a population with a doubling time of
has an initial size of 100. How long is it before the population reaches
10,000?
To answer, we might reason as follows:
 At time t, the number of doublings that have taken place
is
.
 At that time, the size of the population is
.
 If we want to know when the population has reached 10,000, we need
to find a value of x for which
.
 Once we find x, we can find t since
.
The third step is the difficult one: it asks that we
Solve
for x.
The difficulty in solving for x is that we must somehow "undo" the
process of exponentiation. Logarithms are exactly the tool
we need for doing this.
Logarithms to the rescue
For your consideration:
 Explain why the answer to our question on the top of the page
is
.
 Use the demonstration above to estimate
.

How can we compute the logarithm?
We haven't yet told you how to compute the logarithm so it might
not seem like a useful device. Don't worry: we'll learn eventually.
For now, it's safe to trust your calculator.
We can, however, build the graph of the logarithm by dragging the red
ball around again in the picture below.

For your consideration:
In all of the questions below, you should use only information we have
already gathered about the graph of
to make predictions
about
. No previous familiarity with logarithms is
needed nor should it be used.

Are you wondering about the base?
You may wonder what is special about the number 2; that is, why
couldn't we consider a different base like
?
In fact, if you have studied calculus before, you may be wondering when the
magical base "e" will appear, and why logarithms were introduced with
base 2. As a famous detective once said:
"Patience, Watson, all will be soon be clear."
Our motivation in introducing exponential growth as a doubling
process is that it is relatively easy to
multiply factors of 2 rather than other bases (like 7, or
11, or 2.73). It is also very natural to think of this kind of
growth in the simple biological example of cell division.
As we shall see, the "natural"
base e emerges only when we talk about derivatives of the exponential
function. Until then, we may as well discuss any other base.
The demonstration below shows what happens when we vary the base of the
exponential function. The height of the ball above the x axis
represents the base a. As you raise and lower the ball, you change
the base of the exponential.
For your consideration:
 What do you think happens if a=1? Is there a problem with one
or with both of the functions?
 What happens as a is increased ?
 Why should we not try to use negative values of a as base?


