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5 Nonlinear relationships
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10&10a

5.02 The exponential function

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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.

\displaystyle 4^{x}\displaystyle =\displaystyle 4^{8}Write the equation
\displaystyle x\displaystyle =\displaystyle 8Equate the indices

Example 2

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

Worked Solution
Create a strategy

Use the negative index law.

Apply the idea
\displaystyle 8^x\displaystyle =\displaystyle \dfrac{1}{8^2}Write the equation
\displaystyle 8^x\displaystyle =\displaystyle 8^{-2}Use the negative index rule
\displaystyle x\displaystyle =\displaystyle -2Equate the indices

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.

\displaystyle 25^{x+1}\displaystyle =\displaystyle 125^{3x-4}Write the equation
\displaystyle \left(5^2\right)^{x+1}\displaystyle =\displaystyle \left(5^3\right)^{3x-4}Write the bases as powers of 5
\displaystyle 5^{2x+2}\displaystyle =\displaystyle 5^{9x-12}Use the power of a power rule
\displaystyle 2x+2\displaystyle =\displaystyle 9x-12Equate the powers
\displaystyle 2\displaystyle =\displaystyle 7x-12Subtract 2x from both sides
\displaystyle 14\displaystyle =\displaystyle 7xAdd 12 to both sides
\displaystyle x\displaystyle =\displaystyle 2Divide both sides by 7
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

ACMNA239

Explore the connection between algebraic and graphical representations of relations such as simple quadratics, circles and exponentials using digital technology as appropriate

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