Exponential functions

When you hear the phrase, during an epidemic, the number of cases is increasing exponentially. Or perhaps, in connection with an earthquake, someone says it was a 9 on the Richter scale, which is 10 times worse than an 8. How does that work? In order to understand these things, you need to be familiar with the two types of functions, the very important types of functions, that we’re going to be discussing this week, the exponential and logarithmic functions. Now the story begins with simple, humble power functions of the type we know already. A power function means a function whose defining formula is x to the power q, for a fixed q. So x maps to x to the q.

We know a number of such functions already rather well. For example, when q is equal to 1, then the function is simply the identity function. The function that sends x to x. Its graph is a linear function– is a line, as we know. And here it is. Another example we know well is the quadratic function, x-squared. Its graph corresponds to the familiar, parabolic shape. A third example is the function x to the power 1/2. That’s the same as the square root function. Now, it’s a little different from the preceding two, in that it’s not defined for all x’s, but just for positive values of x. And we know its graph.

And as another example consider the reciprocal function, x to the power minus 1, or 1 over x. That’s defined for all x’s that are different from 0. And its graph has a rather hyperbolic look to it, which is something I’ll explain later. Now, here’s a remark. If a is a fixed number, we have defined a to the power q only when q is a rational number, that is a number of the form, plus or minus, m over n, where m and n are positive integers. It’s rather easy to define powers for integer q. And we did it for rational q’s in the following way.

We defined a to the power m over n to be the n-th route or radical of a to the power m. Now, in the expression, let me remind you of some terminology. In a to the q, a is called the base, and q is called the power or the exponent. It is a fact that it’s possible to extend this definition to all real values of the number q, and not just the rational numbers, q. I won’t go into great detail about how this is done. Let me just illustrate by an example. Suppose you wanted to define a to the power square root of 2.

Now as you know– it’s a famous classical fact– the square root of 2 is not a rational number. It cannot be written in the form m over n. However, like any real number, the square root of 2 admits a decimal expansion of arbitrary length. And the fundamental idea here is if you took a certain number of terms in that expansion– for example, if you approximate root 2 by just the five first terms, 1.4142, you can define a to that power, because that’s a rational number. And you would imagine that this would be a good approximation of a to the power root 2. And in fact that’s the idea that can be exploited.

As you take more and more terms in the decimal expansion of root 2, and you take a to those powers, you generate approximations which, in the limit, converge to or approach a certain natural value of a to the root 2. Now this can only be done rigorously when you have the limiting concept, which you always do when you study calculus. But we’ll admit that it can be done, and therefore we’ll admit that there is a reasonable and natural way to define a to the q for any real number q. And some good news about this extended definition is that the extended function you have continues to satisfy all the familiar rules for power functions that we already know and love.

And here they are. So visually everything looks the same, and there’s really nothing to worry about. Now, the consequence of all this is we can consider power of functions, x to the q, for any fixed power q in r. What do these functions look like? Well they look pretty much like they did for rational values of q. For example, if q is a positive number, then this defines a strictly increasing function of x. Remember that x has to be positive for these power functions. And if q is 1, it’s a linear function. And if q is negative, then you get a decreasing function of x.

And they all have the value 1, when x is equal 1, because 1 to the q equals 1. All right, now that was our review of power functions. And in a power function the q is fixed– the exponent, the power– and x is the variable. We are now going to talk about exponential functions. So this is new. An exponential function is a function which takes a point x and returns the value a to the power x. So you see now that it’s the base, a, which is fixed, and it’s the exponent, x, which is the variable. That’s an exponential function. We’re going to study the behaviour of such functions.

Personally, I always feel better when I know the graph of a function, at least vaguely. I feel I know what the function looks like when I am familiar with its graph. When a is greater than 1, it’s not hard to see and even to prove that the function, a to the x, is a strictly increasing function. And its graph looks like this. You will note that the graph goes through the point 0, 1. Why? Because 0– because a to the power of 0 is equal to 1. Now, you could prove rigorously that the function is strictly increasing. It’s not difficult. The proof goes as follows. You take a y greater than x.

You look at the difference, a to the y minus a to the x. And using some familiar laws of arithmetic, you express that difference as the product of two positive numbers. And hence it’s positive, and that’s the end of the proof. When a is strictly less than 1 but still positive now– so we want a between 0 and 1– then the function a to the x is a decreasing function. And it still goes through the point 0, 1. It is a fact that exponential growth, or– contrariwise– exponential decay, is much faster than it is in power functions. Let me show you what I mean.

We’ll compare the two functions, x goes to x-squared– a quadratic power function– and the function x goes to 2 to the power x– an exponential function with base 2. Here are some graphs where the x is between minus 10 and 10. As you can see, the graph of the exponential function on the left, when x is negative, goes down to the x-axis, goes down to 0, awfully fast, even when x has relatively small but negative values. On the other hand, for positive values of x, if you compare the quadratic and the exponential functions, you might think that they are pretty much of the same growth. But that’s really just an illusion.

As soon as you let x get a little bigger– like, for example, when x is as big as 10 in this graph– you’ll see that by the time x is 10, the graph of the exponential function, the blue curve here– it’s rising so quickly that the graph looks almost vertical. Exponential growth is eventually, at least, really very fast growth indeed.