Chapter 15Deeper Exploration of Logs andExponentials1Not all of the models that we can use to describe real world data are based on power func-tions or polynomials. In fact, we saw in earlier chapters that there are many situations whereexponential or logarithmic models may be needed. We also developed a way of interpretingthe coefficients of these models using parameter analysis. However, parameter analysis doesnot give us the power needed to locate maxima and minima for such models. Only calcu-lus tools, specifically the derivative, can do this. In this chapter, you will work with thederivatives of exponential and logarithmic functions, and you will further apply these toolsto analyze models of the business world. When you have finished this chapter, you will knowhow to deal with many of the basic functions found in the real world. The symbolic analysisportion of this chapter will show you how, using multiplication, division and composition ofmodels, we can build many more types of models and analyze them using calculus.• As a result of this chapter, students will learn√How to use the calculus tool of derivatives to analyze models involving logarithms√How to use derivatives to analyze models involving exponentials√How compound interest works, including continuously compounded interest• As a result of this chapter, students will be able to√Take derivatives of exponential functions√Take derivatives of logarithmic functions√Compute compound interest1c2011 Kris H. Green and W. Allen Emerson443444 CHAPTER 15. LOGARITHMIC AND EXPONENTIAL MODELS15.1 Logarithms and their derivativesAs we have seen, there are many times when the model you develop will need to go beyondthe power or polynomial models. For a multitude of reasons, the exponential and logarithmicmodels are the next most common models:1. Exponentials are easy to interpret based on percent changes; thus, they can easilyrepresent 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 modelingdata, specifically the problem of non-constant variance.3. Logarithms can be useful for simplifying many other models for analysis, since loga-rithms (remember the properties listed in section 12.2.1) can be used to convert manyexpressions involving multiplication and division into addition and subtraction prob-lems.These reasons alone are sufficient to justify learning how to properly use derivatives toanalyze such functions. Before we get to technical, though, it’s worth looking at the functionsthemselves and trying to figure out what we expect to happen. If we look at a graph of anexponential function, we notice immediately that the slope is always increasing. The slopeis 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 alot, we have to remember that we are only looking at a small portion of the complete graphof the function, so it is possible that somewhere far from where we are looking this behaviorwill 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 functionshows 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 functionseems to be concave down everywhere, even though it is increasing. Is it possible thatsomewhere far down the line the graph actually starts to decrease? We must also bear inmind that whatever we learn about one of the functions can be applied to the other, sincelogarithms 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 derivativeformulas for the power and polynomial models. Along the way we will encounter some otherrules for taking derivatives: the chain rule, product rule and quotient rule. These will give usthe ability to differentiate (take the derivative of) functions that are made of combinationsof basic functions like logarithms and power functions. The next section will explore theexponential function and its applications to one of the most frequently used economics andbusiness scenarios: compound interest.15.1. LOGARITHMS AND THEIR DERIVATIVES 44515.1.1 Definitions and FormulasComposition of Functions This is one way of making a new function from two old func-tions. Essentially, we take one function and ”plug it into” the other function. Forexample, if we compose f(x) = 2x3and g(x) = 4x−5 we get either h(x) = (f ◦g)(x) =f(g(x)) = 2(4x −5)3or we get k(x) = (g ◦ f)(x) = 4(2x3) − 5 depending on the orderof the composition. In general, the two orders are not the same.Chain rule We’ll be using this rule a lot. The symbolic analysis section will explain it inmore detail, but the basic idea is that if you have a function composed with anotherfunction and you need the derivative of the combined object, you use the chain ruleto ”chain together” derivatives of each function. For example, if we start with thefunctions f (x) and g(x) above and compose them into h(x) the new function h is nolonger a simple power function or polynomial (although we could multiply it out intoa polynomial.) But since it is composed of these simpler functions, we can still takeit’s derivative. In fact, the chain rule says thatddxf(g(x)) =dfdg·dgdx.Thus h0(x) = [df /dg][dg/dx] = [2 · 3g(x)2] · [4] = 24(4x − 5)2. A derivation and proofof the chain rule are somewhat technical; for now, think of this as a way of chainingtogether the derivatives so the objects which look like (but aren’t really) fractions willcancel out. In the above illustration of the chain rule, the first ”fraction” has thenumerator we want (df) and the second ”fraction” has the denominator we want (dx).Each of these ”fractions” has a dg term that ”cancels out” to give the derivative wewant: df/dx.Product rule The product rule allows us to take derivatives of functions that are productsof simpler functions. It says thatddx[f(x) · g(x)] = g(x) ·dfdx+ f(x) ·dgdx.The proof of