Exponential and logarithmic functions illuminated.

### What are exponential and logarithmic functions?

**exponential function**is a function of the form where is a positive real number. The domain of an exponential function is and the range is .

**logarithmic function**is a function defined as follows where is a positive real number. The domain of a logarithmic function is and the range is .

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 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 .

#### Properties of logarithms

Let be a positive real number with .

### 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.

From here, we can solve for directly.

Our equation is really a quadratic equation in ! The left-hand side factors as , so we are dealing with For the first:

From the second: . Look back at the graph of above. What was the range of the exponential function? It didn’t include any negative numbers, so has no solutions.

The solution to is .

### Logarithmic equations

Let’s use quadratic formula to solve this.

What happens if we try to plug into the equation? Both and are negative. That means, the logarithms of these values is not defined.

It turns out that is a solution of the equation , but not a solution of the original equation .

When working with logarithmic equations, we must always check that the solutions we find actually satisfy the original equation.

The only solution is .