Exponential and logarithmic functions illuminated.

Exponential and logarithmic functions may seem somewhat esoteric at first, but they model many phenomena in the real-world.

What are exponential and logarithmic functions?

Is an exponential function?
yes no

In either definition above is called the base.

Remember that with exponential and logarithmic functions, there is one very special base: This is an irrational number that you will see frequently. The exponential with base , is often called the ‘natural exponential’ function. For the logarithm with base , we have a special notation, is ‘natural logarithm’ function. We’ll talk about where comes from when we talk about derivatives.

Connections between exponential functions and logarithms

Let be a positive real number with .

  • for all positive
  • for all real
What exponent makes the following expression true?

What can the graphs look like?

Graphs of exponential functions

Graphs of logarithmic functions

Properties of exponential functions and logarithms

Working with exponential and logarithmic functions is often simplified by applying properties of these functions. These properties will make appearances throughout our work.

Properties of exponents

Let be a positive real number with .

What exponent makes the following true?

Properties of logarithms

Let be a positive real number with .

What value makes the following expression true?
What makes the following expression true?

Exponential equations

Let’s look into solving equations involving these functions. We’ll start with a straightforward example.

Of course, if we couldn’t rewrite both sides with the same base, we can still use the properties of logarithms to solve.

Solve the equation: .
The equation has no solutions.

Logarithmic equations