Natural logarithms

We’re now going to define the famous logarithmic functions. The key to doing this– we’re going to look at the natural exponential function that we already know, and we’re going to consider its inverse. Now, as you know, the function e to the x as a mapping from the real numbers to the open interval 0 infinity is bijective. The fact that it’s injective, that it assumes any given value at most once– and in fact, exactly once– is due to the fact that it’s strictly increasing. That’s easy to see. The fact that any positive value, strictly positive, can be attained by the function e to the x we’re going to admit.

Graphically, it’s pretty clear when you look at the graph of the natural exponential function. Well, given that it’s bijective, we are then able to apply the previous theory of inverse functions and say that it admits an inverse function. And we’re going to give that a name. We’re going to call it the natural logarithm, abbreviated quite often by the letters ln. And this natural logarithm, then, will be a function that maps in the other direction. It’s going to be defined on the open interval 0 infinity, and its values are going to be in R. So to summarize, y is equal to ln x if and only if e to the y is equal to x.

In this expression, the x has to be a strictly positive number, and the y can be anything in the real line. What is the graph of the natural logarithm function going to look like? Well, we already have it, basically. Because we know that the graph of the inverse function is obtained from the original function’s graph by reflecting about the line y equals x. So there we have it. Let’s nonetheless note explicitly a few facts about this natural logarithm. To begin with, it’s equal to 0 at x equals 1. Why? Because e to the 0 is equal to 1. We notice that ln is a strictly increasing function.

It’s negative precisely when the argument x is between 0 and 1, and it’s positive when x is to the right of 1. Also, the function ln has a vertical asymptote at x equals 0. That means that it plummets down to minus infinity as x approaches 0 from the right-hand side. And finally, let’s note that as x goes off to plus infinity off to the right, the function lnx does become unbounded– rather slowly, but it’s unbounded. Now, because ln is the inverse of the exponential function, the effect of each function is to cancel out the other. So the ln of e to the y is y, and the exponential of ln x is x.

Here’s a very important fact about logarithms– namely, that they convert multiplication into addition. More precisely, if you have two positive numbers u and v, then ln of the product u v is ln u plus ln v. Proof of that is very easy to give. Let’s see how it works. By definition of inverse function, what does ln of u v mean? It means the unique real number– let’s temporarily call it r– having the property that e to the r is u v. So what we need to show is that that number r is ln u plus ln v. That is, we need to show that the exponential of that sum is equal to u v.

But that follows from simple properties of exponential that we’re familiar with and the fact that the exponential of ln u is u, for example. And there is the proof. There are other rules of the function ln that are proven just as easily– for example, a quotient rule. ln of u over v is ln u minus ln v. As a corollary of that formula, by taking u equals 1 and remembering that ln 1 is 0, we get that ln of the reciprocal, ln of 1 over v, is given by minus ln v. Finally, a power rule for the natural logarithmic function. ln of u to the power r is equal to r ln u.

The power r can come out in front as a multiplicative factor. Now, take care in using that last rule. It is certainly true that ln of root x is 1/2 lnx. Why so? Because root x ix x to the power 1/2. And this is simply, then, the power rule for ln in the case r equals 1/2. It is not true, however, that the square root of lnx is 1/2 lnx. It looks similar, but it’s quite different. And in fact, this is false. So beware. Here’s a simple example of how we can use these rules to develop or expand or simplify certain complicated expressions like the one you see before you.

I first apply the ln law for quotients to write it as ln of the numerator minus ln of the denominator. Then I use the product law to separate the a and the b cubed into two lns. And now I use the power law to bring the 3 from the b cubed in front and the 1/2 from the c to the 1/2 in front, and you’ll notice that I’ve now simplified my expression into basic terms. It turns out that logarithms can be defined to not just the base e, but to an arbitrary base a, provided it’s positive and different from 1. Here’s the definition.

The function log a, or log to base a, or log with base a, is defined as follows. Log a of x means the unique number y that satisfies the equation a to the y equals x. Therefore, you see that ln– our function from before, the natural logarithm– is simply the special case of this general logarithmic function in which the base a has been taken equal to the Euler number e. So to summarize, to say that y is log a of x is equivalent to saying that a to the y equals x. Now we have a simple fact here that relates the logarithm to any base a to the natural logarithm.

So in a sense, all the study of these other log functions can be reduced to things we know about the natural logarithm. Proving that proposition is not difficult. Let’s begin by writing a to the power log a x equals x. That’s the definition of log a. I now take the ln of both sides, and I use the power rule on the left so that the exponent, log a of x, comes out in the front as a multiplicative factor. Then, in the last equation, it suffices to divide both sides by ln a, and you get exactly the formula affirmed in the proposition.

From this fact, from this proposition, you can derive formulas that show that the log a function satisfies all the same rules that ln did. For example, log a of a product u v is the sum of log a u plus log a v, and so on for quotients and powers. Now, there are two special base values for logarithmic functions that are of greatest interest. One of them is e– the natural log, of course– and the other one is 10. Why 10, you say? Well, it’s because we have 10 fingers. I’ll explain that very soon.