Section 1.5: The Exponential Function

The exponential function

For a fixed real number a the function

is called the exponential function with base a. Our first goal today is to understand what this function is and study some of its characteristics.

If x is a positive integer n, the exponential function is easy to understand.

To get a better understanding of a^x in the case when x is not a positive integer, we have to spend some time thinking about the exponentiation operation and its properties. There are three further extensions to this definition that you can take as given:

These ideas, when taken together allow us to define the exponential function a^x in the case where x is a rational number.

Further facts about exponentials

Assuming that a and b are positive real numbers, the following facts hold for exponentials

These facts are all easy to prove when x and y are positive integers, somewhat more challenging but still possible to prove when x and/or y are rational numbers, and almost impossible to prove (without calculus) when either x or y are irrational numbers. Nonetheless, we will take these as given. (We have to start somewhere.)

The function 2^x

To summarize what we have learned so far, and to make things a little more concrete for the moment, consider the special case a = 2. Here is a table of some of the values of 2^x for various values of x.

x 2^x
1 2
2 4
3 8
1/2 1.414213
3/2 2.828427
4/3 2.519842
-1 0.500000
-2 0.250000
-1/2 0.707106
-3/2 0.353553

x	2^x
1	2
2	4
3	8
1/2	1.414213
3/2	2.828427
4/3	2.519842
-1	0.500000
-2	0.250000
-1/2	0.707106
-3/2	0.353553

Plotting all of these points and connecting the dots causes the shape of the curve y = 2^x to start to emerge.

2^x for irrational x

Suppose we wanted to compute, say, . How would we do that? Nothing we have seen so far can help us, because the best we can do is compute 2^x for rational x. is an irrational number - you can't write it as a fraction p/q. The next best thing you can do is to find a sequence of rational numbers that get ever closer to and see what happens when you plug those rational numbers into the exponential.

The table below shows one such sequence. As you can clearly see from the table, as the rational numbers get ever closer to the limiting value of the corresponding values of the exponential function also appear to be approaching a limiting value.

x = p/q 2^p/q
1 1 2
1.5 2.828427124746190
1.416666666666666 2.669679708340069
1.414215686274509 2.665148066255112
1.414213562374689 2.665144142675931
1.414213562373095 2.665144142690225

	x = p/q	2^p/q
1	1	2
	1.5	2.828427124746190
	1.416666666666666	2.669679708340069
	1.414215686274509	2.665148066255112
	1.414213562374689	2.665144142675931
	1.414213562373095	2.665144142690225

By using a similar limiting process you can define the value of 2^x for any x, rational or irrational. Here is the plot of the exponential function y = 2^x.

Ultimately, there is nothing special about 2 as the value of a. Changing the base a leads to a whole family of different exponential curves. One value of a has something special going for it. The plot below shows a plot of a^x along with a plot of the curve y = x + 1, which passes through (0,1) and has a slope of one. When a is approximately equal to 2.71, the exponential curve appears to just touch the line.

The precise value of a that makes the exponential curve perfectly tangent to the blue line is

e = 2.718281828459045

The exponential function that uses this number as its base, e^x, is called the natural exponential function (or often just the exponential function for short).

Section 1.6: Inverse Functions and Logarithms

A function is said to be one-to-one if for every y in the range of f there is exactly one x in the domain for which f(x) = y. Below are two examples. The function f(x) = x² (shown in blue) is not one-to-one, while the function f(x) = x³ (shown in green) is.

For one-to-one functions we can define an inverse function. If the function is one-to-one we can solve the equation

for x in terms of y. The result is a new function g satisfying the equation

x = g(y)

Switching the roles of x and y gives us the inverse function for f, commonly written f^-1.

Here is a concrete example. Let f(x) = x³. f is one-to-one, so we should be able to compute its inverse function.

So we have

An easy way to visualize an inverse function is to take the graph of the original function and flip it about the line y = x.

A better definition for the inverse function

Although the 'solve for x' method and the 'flip about the line y = x' methods are fairly useful for understanding what the inverse function is, they are not sufficiently precise to use in proofs. A better definition for inverse function is the following.

Definition If f(x) is a one-to-one function defined on some domain, the inverse function of f(x), f ^-1(x) is the function with the properties that for every x in the domain of f and for every x in the range of f.

The Natural Logarithm Function

The natural exponential function is another example of a one-to-one function. Its inverse function is called the natural logarithm function, written ln x.

Some facts about ln x

Using the fact that ln x is the inverse function of e^x quickly leads to the following:

The key to proving these is to begin with the observation that since the natural exponential and the natural logarithm are inverse functions of each other we immediately have