Exponents and logarithms: inverse functions

Let's begin with a concrete question.

Suppose that a population with a doubling time of $ \tau $ 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 $ x=t/\tau $ .
  • At that time, the size of the population is $ 100\cdot 2^x $ .
  • If we want to know when the population has reached 10,000, we need to find a value of x for which $ 10,000 = 100 \cdot 2^x $ .
  • Once we find x, we can find t since $ t=x \tau $ .

The third step is the difficult one: it asks that we

Solve $ 100= 2^x $ 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

To solve our problem, we will introduce a new function called the logarithm. More specifically, since we are dealing here with base 2 exponentials, we will define the base 2 logarithm like: if we have a positive real number $ c $ , the base 2 logarithm is the number


\[ b \mbox{ which satisfies } 2^b = c. \]

We will call $ b $ the base 2 logarithm of $ c $ and denote it by $ b = \log_2(c) $ .


That sounds confusing, but it can help to see the idea on a graph. If you move the red ball up and down, you will change the number $ c $ which is represented by the length of the vertical arrow. You can then see the logarithm $ log_2(c) $ as represented by the length of the horizontal arrow. As it should be, the logarithm of $ c $ is the number which, when exponentiated, gives $ c $ back.


For your consideration:

  1. Explain why the answer to our question on the top of the page is $ x = \log_2(100) $ .

  2. Use the demonstration above to estimate $ x = \log_2(100) $ .
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 $ 2^x $ to make predictions about $ log_2(x) $ . No previous familiarity with logarithms is needed nor should it be used.

  • Explain why $ log_2(x) $ is not defined for negative values of x.

  • Explain why it must be true that the value of $ log_2(x) $ gets infinitely large when x approaches zero.

  • Show that $ log_2(1)=0 $ .

  • Explain why it must be true that the rate of increase of the function $ y=log_2(x) $ is slower than that of the straight line $ y=x $ when $ x > 1 $ .



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 $ 3^x $ ? 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?