15.1 Logarithms and their derivatives

As we have seen, there are many times when the model you develop will need to go beyond the power or polynomial models. For a multitude of reasons, the exponential and logarithmic models are the next most common models:

1. Exponentials are easy to interpret based on percent changes; thus, they can easily represent mathematically the process of accruing interest for loans or other accounting-related phenomena.
2. Logarithms are useful for dealing with some of the potential problems in modeling data, specifically the problem of non-constant variance.
3. Logarithms can be useful for simplifying many other models for analysis, since logarithms (remember the properties listed in section 12.2.1) can be used to convert many expressions involving multiplication and division into addition and subtraction problems.

These reasons alone are sufficient to justify learning how to properly use derivatives to analyze such functions. Before we get to technical, though, it’s worth looking at the functions themselves and trying to figure out what we expect to happen. If we look at a graph of an exponential function, we notice immediately that the slope is always increasing. The slope is always positive, and the curve is always concave up. Thus, we expect the derivative to (a) always be positive and (b) increase as x increases. While these observations seem to tell us a lot, we have to remember that we are only looking at a small portion of the complete graph of the function, so it is possible that somewhere far from where we are looking this behavior will change. Once we have the derivative in hand, however, we can find out if this happens. (You’ll have a chance to work with this in one of the problems at the end of the chapter.)

This is all in stark contrast to logarithmic functions. The graph of a logarithmic function shows more complex behavior. While it is true that the graphs seems to be always increasing, notice that the slope is decreasing as we move to the right. Thus, the logarithmic function seems to be concave down everywhere, even though it is increasing. Is it possible that somewhere far down the line the graph actually starts to decrease? We must also bear in mind that whatever we learn about one of the functions can be applied to the other, since logarithms and exponentials are inverses of each other.

The following section is devoted to learning about the derivatives of logarithmic functions. The development of this will mimic the path we took in chapter 14 to develop the derivative formulas for the power and polynomial models. Along the way we will encounter some other rules for taking derivatives: the chain rule, product rule and quotient rule. These will give us the ability to differentiate (take the derivative of) functions that are made of combinations of basic functions like logarithms and power functions. The next section will explore the exponential function and its applications to one of the most frequently used economics and business scenarios: compound interest.