EXAMPLE 1 Solve by equating exponents

Rewrite 4 and as powers with base 2.

12

Solve 4 = x 1

2

x – 3

(2 ) = (2 )2 x – 3x – 1

2 = 2 2 x – x + 3

2x = – x + 3

x = 1

SOLUTION

4 =x 1

2

x – 3

Write original equation.

Power of a power property

Property of equality for exponential equations

Solve for x.

The solution is 1.ANSWER

EXAMPLE 1 Solve by equating exponents

Check: Check the solution by substituting it into the original equation.

Substitute 1 for x.

Simplify.

Solution checks.

4 =?12

– 2

1 – 3

4 =1 ? 12

4 = 4

GUIDED PRACTICE for Example 1

Solve the equation.

1. 9 = 27 2x x – 1

9 = 27 2x x – 1

SOLUTION

(3 ) = (3 )2 x – 12x 3

3 = 3 4x 3x – 3

Rewrite 9and 27 as powers with base 3.

Write original equation.

Power of a power property

4x = 3x –3 Property of equality for exponential equations

9 = 27 2x

9 = (9•3) 2x x – 1

GUIDED PRACTICE for Example 1

4x – 3x = –3

= –3 Solve for x.

Property of equality for exponential equations

The solution is – 3.ANSWER

GUIDED PRACTICE for Example 1

Solve the equation.

2. 100 = 1000

7x + 1 3x – 2

SOLUTION

(10 ) = (10 )2 3x – 27x+1 3 Rewrite 100and 1000 as powers with base 10.

Write original equation.

Power of a power property

14x + 2 = 9x – 6 Property of equality for exponential equations

100 = 1000 7x + 1 3x – 2

(10 ) = (10 )9x – 614x+2

GUIDED PRACTICE for Example 1

14x – 9x = – 6 – 2

Solve for x.

Property of equality for exponential equations

5x = – 8 Property of equality for exponential equations

– 8 5

The solution is .ANSWER

x = – 8 5

GUIDED PRACTICE for Example 1

Solve the equation.

81 = 3 – x 1

3

5x – 6

3 = 3 12 – 4x – 5x+6

Rewrite 81 and 1/3 as powers with base 3.

Write original equation.

Power of a power property

12 – 4x = – 5x+6 Property of equality for exponential equations

(3 ) = (3 )

3 – x4 – 1 5x – 6

SOLUTION

3. 81 = 3 – x 13

5x – 6

GUIDED PRACTICE for Example 1

– 4x + 5x = 6 – 12 Property of equality for exponential equations

x = – 6 Solve for x.

The solution is – 6 .ANSWER