Unit 3 of 10

Exponential and Logarithmic Functions

  • 11 lessons
  • Parts 71 to 92

Exponential functions describe quantities that grow or shrink by the same factor over equal steps, such as populations, savings and radioactive material. Logarithms answer the reverse question of which exponent gives a certain value, so the two families are studied side by side.

This unit introduces the number e, the graphs and properties of logarithms, and methods for solving exponential and logarithmic equations. It closes with growth and decay models, choosing a model that fits data, and logistic growth, where a quantity levels off at a carrying capacity.

Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Exponentials, e, and Logarithms

  1. Exponential Functions and Growth/Decay

    What an exponential function is, why its base must be positive and not 1, how equal steps in the input give equal ratios in the output, and how growth and decay differ in graph, range and asymptote.

    Theory part 71Practice part 72

  2. The Natural Base e

    Where the number e comes from, how compounding interest more and more often leads to it and to continuous growth, and why a table of values suggests a limit but cannot prove it.

    Theory part 73Practice part 74

  3. Logarithmic Functions as Inverses

    What a logarithm is, why its base must be positive and not 1 and its input positive, how it undoes an exponential, and the domain and range of common and natural logarithms.

    Theory part 75Practice part 76

  4. Graphs of Exponential and Logarithmic Functions

    The shapes of exponential and logarithmic graphs, why they are mirror images, and how reflections and shifts move their key points, their asymptotes, their domains and their ranges.

    Theory part 77Practice part 78

Properties of Logarithms

  1. Product and Quotient Properties of Logarithms

    Why the logarithm of a product is a sum and the logarithm of a quotient is a difference, when these rules hold, and how to expand and condense logarithms without changing the domain by accident.

    Theory part 79Practice part 80

  2. The Power Rule and Change of Base for Logarithms

    How an exponent inside a logarithm moves out in front as a factor, why that needs a positive argument, and how to rewrite any logarithm in base 10 or base e, exactly or as a decimal.

    Theory part 81Practice part 82

Exponential and Logarithmic Equations

  1. Solving Exponential Equations

    How to solve equations with the unknown in an exponent, by matching bases or by isolating the power and taking logarithms, and how to state the answer exactly before rounding it.

    Theory part 83Practice part 84

  2. Solving Logarithmic Equations

    How to solve equations with the unknown inside a logarithm by rewriting them in exponential form, and why every candidate has to be checked against the domain of the original equation.

    Theory part 85Practice part 86

Models

  1. Exponential Growth and Decay Models

    How to build and read exponential models from an initial value and a growth factor, how discrete and continuous rates compare, where half-life and doubling time come from, and when such a model stops being believable.

    Theory part 87Practice part 88

  2. Choosing and Interpreting Function Models

    How to read the variables, parameters and units of a model, compare a linear and an exponential fit with residuals, and judge when an estimate from data deserves trust.

    Theory part 89Practice part 90

  3. Logistic Growth and Model Limits

    How the logistic function models growth that starts out nearly exponential and levels off at a carrying capacity, what each parameter means, and where the model stops matching reality.

    Theory part 91Practice part 92