The Exponential Function and Logarithms Chapter 11

11.1 Introduction Motivating example. Suppose you invest a USD with a bank at an annual interest rate of r percent for x years. We know that at the end of year x the value of your deposit will be given by Try to answer the following question: in how many years will your principal a double? In other words, we want to solve the following equation: However, this is where we get stuck: given r and x, we can compute the right-hand side, but not the other way round. Given the right-hand side, we cannot compute x. In fact, to solve this equation we need two new concepts: Exponential function Inverse function

11.2 Exponential Function An example of exponential function: Number 10 in the formula above is called the base. The base can be any number strictly greater than 1. Key features of exponential function: Always strictly increasing function of its argument Never cuts the x-axis Vertical intercept is at y=1 for any base Increases very fast with increases in argument x (grows at an increasing rate) Exponential function is always returning strictly positive values: y>0

11.3 Logarithmic Function as Inverse to Exponential Function Definition. Consider a function . Suppose that for each y there exists only one value of x such that . In this case we say that there exists an inverse function that maps y into x, written as . Why is it essential that there be only one single x for each y?

Example 11.1

Logarithmic Function Since the exponential function satisfies the conditions for the existence of an inverse, the inverse function exists (see Graph b). The problem is, how do we call this function? Definition. Given the exponential function , the inverse function is defined as read as “logarithm of y to base 10.” Note 1. Logarithms to base 10 are called common logarithms. Note 2. Logarithms are often called “logs.”

11.4—11.6 Graph and Properties of Logarithmic Function Key features. The logarithmic function is increasing everywhere The logarithmic function is defined for x>0 The logarithmic function cuts the x axis at x=1 When x is between zero and one, the logarithmic function is negative. The logarithmic function grows at a decreasing rate

Rules for Logs In general, logarithms simplify arithmetic operations: Rule 1. Product to sum: Rule 2. Power to product: Rule 3. Ratio to difference: Simple rules:

11.8 Using Logs Remember our motivating example: we want to know in how many years the value of our account will double. This problem boils down to the following equation: Take logs of both sides of the equation: For instance, if r=3%, it will take 23 years for your account to double.

11.9 More Exponential Functions Consider a function where a>0 is a parameter. In case a>1, the graph of is made steeper: If 0<a<1, the graph of is made flatter:

Case of Negative a Consider a function where a<0. For instance, if a=-1, we get a symmetric graph to with respect to the y axis:

Generalizing Exponents and Logarithms In the formula for the exponential or logarithmic functions, 10 does not have to be the only base. Consider, for instance, . It has the same properties as , but the graph looks a little bit different. The inverse function to is . Later we shall see that it is very convenient to stick to base called Euler’s number. e=2.7182818284590452353602874713527...