# 5.02 The exponential function

Lesson

## The exponential function

An exponential expression is an expression of the form A^x where A is a positive number and x is a pronumeral. A is called the base of the exponential. An exponential equation is an equation where one or both sides are exponential expressions. To solve an exponential equation, we can write both sides of the equation as exponentials with the same base and then the indices will be equal.

### Examples

#### Example 1

Solve 4^{x}=4^{8}.

Worked Solution
Create a strategy

Equate the indices and solve for x.

Apply the idea

Both sides of the equation are exponentials with the same base. So it follows that the powers must be equal.

#### Example 2

Solve 8^{x}=\dfrac{1}{8^2}.

Worked Solution
Create a strategy

Use the negative index law.

Apply the idea

#### Example 3

Solve 25^{x+1}=125^{3x-4}.

Worked Solution
Create a strategy

Write both sides as indexes with the same base.

Apply the idea

25 and 125 are both powers of 5 since 5^2=25 and 5^3=125. So we can write both sides with a base of 5.

Idea summary

An exponential expression is an expression of the form A^x where A is a positive number and x is a pronumeral.

A is called the base of the exponential.

An exponential equation is an equation where one or both sides are exponential expressions.

To solve an exponential equation, we can write both sides of the equation as exponentials with the same base and then the indices will be equal.

### Outcomes

#### VCMNA339

Explore the connection between algebraic and graphical representations of relations such as simple quadratic, reciprocal, circle and exponential, using digital technology as appropriate