Chapter 10
Exponential and Logarithmic Functions
One rumor reaches two people, each of them tells two more, and by the tenth round the count has run away from you. Nothing was added at any step; the amount was simply multiplied again and again. The straight lines and steady rates you already know cannot describe that, and they cannot answer the reverse question either: given how large the crowd has grown, how many rounds did it take? Exponential functions are the ones built for growth that multiplies. What has to be true of a function before it can behave that way?
What You'll Explore
6 lessons.
- Exponential Functions and Graphs
Suppose every step forward multiplies the amount by the same factor instead of adding the same amount. What kind of function behaves that way, and can any number serve as its base? This lesson settles both questions, then looks at the graphs and at how such growth fares against a curve that merely squares its input.
- Introduction to Logarithms
Two raised to some power gives sixteen, and a little trial will find it. Ask instead which power of two gives one thousand, and trial runs out quickly. Here you meet the operation that answers such a question outright, along with a restriction on what you are allowed to ask about.
- Properties of Logarithms
Working with exponents, you learned a small set of rules for combining and rearranging them. Does the operation that undoes them come with its own set? This lesson looks for such rules, checks what they quietly assume, and deals with a calculator that only knows a couple of bases.
- Solving Exponential and Logarithmic Equations
Once an unknown sits in an exponent, the usual moves of adding and dividing cannot reach it. There is now a tool that can, but using it takes some care: certain steps produce answers that look right and fail when you check them. Finding out which ones, and why, is the work here.
- Exponential Growth and Decay
A population rising five percent a year and a sample losing five percent of what remains each year seem like opposite situations. Are they really two stories, or one? This lesson builds the model behind both, separates two things that are easy to confuse, and asks how long a doubling takes.
- The Natural Base e
Every exponential so far has used a base picked to suit the situation, two for doubling or ten to match our number system. Yet one particular base shows up again and again in science and finance, and nobody chose it for convenience. Where does a number like that come from, and what does it make easier?