Exponential and Logarithmic Functions
Every function in the first three chapters was built by adding, multiplying, and dividing powers of the variable. This chapter puts the variable in the exponent instead, and the difference is not a matter of degree: a quantity that grows by a fixed percentage each period eventually outruns any polynomial, which is why the same family models compound interest, bacterial growth, and radioactive decay alike. Inverting that family gives the logarithm — the function that answers “to what exponent?” — and with it the algebra needed to solve for a variable that is stuck in an exponent. The chapter closes by turning the machinery on real data: choosing between exponential, logarithmic, and logistic models, and fitting them by regression.
Sections
- Exponential Functions — identifying exponential functions, evaluating them, finding an equation from two points or from a graph, and applying the compound-interest and continuous-growth formulas.
- Graphs of Exponential Functions — graphing the parent function from a table of values, reading its horizontal asymptote, domain, and range, and building every shift, stretch, compression, and reflection of it.
- Logarithmic Functions — converting between logarithmic and exponential form, evaluating logarithms by hand and with a calculator, and using the common and natural logarithms.
- Graphs of Logarithmic Functions — finding the domain of a logarithmic function, graphing the parent function and its vertical asymptote, and applying shifts, stretches, compressions, and reflections to it.
- Logarithmic Properties — the product, quotient, and power rules, using them together to expand or condense a logarithmic expression, and the change-of-base formula.
- Exponential and Logarithmic Equations — solving exponential equations by matching bases or taking logarithms, solving logarithmic equations with the definition of a logarithm and the one-to-one property, and rejecting extraneous solutions.
- Exponential and Logarithmic Models — modeling growth and decay, Newton’s Law of Cooling, logistic growth, choosing a model that fits a data set, and expressing an exponential model in base .
- Fitting Exponential Models to Data — building exponential, logarithmic, and logistic models by regression, and reading the correlation coefficient as a measure of goodness of fit.
The sections above are adapted from Precalculus 2e, Chapter 4 by Jay Abramson, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org.