
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.

#### 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?

### 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?