Unit 5 of 8

Exponential and Logarithmic Functions

  • 7 lessons
  • Parts 73 to 86

Exponential functions model growth and decay by a constant factor, and logarithms undo them. This unit compares exponential functions with different bases, singles out the natural exponential function, and defines logarithms as inverse functions.

The natural logarithm, the laws of exponents and logarithms with their conditions, and the logarithm of an absolute value follow, since all of them appear when calculus differentiates and integrates. The unit ends with solving exponential and logarithmic equations.

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.

Exponential Functions

  1. Exponential Functions with Different Bases

    How an exponential function multiplies its output by the same factor over every equal input step, how the base decides growth or decay, and how a context narrows the domain.

    Theory part 73Practice part 74

  2. The Natural Exponential Function

    Why e is an exact constant and not a rounded decimal, what the graph of the natural exponential looks like with its exact landmarks, and how constant multiples in front and in the exponent model growth and decay.

    Theory part 75Practice part 76

Logarithms

  1. Logarithms as Inverse Functions

    What a logarithm means as the exponent that produces a given number, why that number must be positive, and how its graph comes from the exponential with the same base by swapping inputs and outputs.

    Theory part 77Practice part 78

  2. The Natural Logarithm Function

    How the natural logarithm of a number is the exponent on e that gives that number, why it needs a positive input but can give negative outputs, and how to find the domain and asymptote of a shifted logarithm.

    Theory part 79Practice part 80

  3. Using Exponent and Logarithm Laws

    How to combine powers of one base and split logarithms into simpler pieces, which conditions each law needs, and how to keep the domain of the original expression while you rewrite.

    Theory part 81Practice part 82

  4. Natural Logarithm of Absolute Value

    Why ln|x| is defined for every nonzero x, how its graph splits into two mirror image branches, and how to find exclusions and zeros when the center shifts.

    Theory part 83Practice part 84

Equations

  1. Solving Exponential and Logarithmic Equations

    How to solve an equation with the unknown in an exponent or inside a logarithm by choosing the right inverse step, and why every candidate must be checked in the original equation.

    Theory part 85Practice part 86