7-5 Logarithmic & Exponential Equations

Terms and Conceptsthese will be on a quiz

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 2x – x + 3

2x = –x + 3

x = 1

SOLUTION

4 =x 12

x – 3

Write original equation.

Power of a power property

Property of equality for exponential equations

Solve for x.

The solution is 1.ANSWER

GUIDED PRACTICE for Example 1

Solve the equation.

1. 9 = 27 2x x – 1

SOLUTION –3

2. 100 = 1000 7x + 1 3x – 2

SOLUTION – 8 5

3. 81 = 3 – x 1

3

5x – 6

SOLUTION –6

EXAMPLE 2 Take a logarithm of each side

Solve 4 = 11.x

4 = 11x

log 4x = log 4 114

log4

x = 11

x = log 11 log 4

x 1.73

SOLUTION

Write original equation.

Take log of each side.4

Change-of-base formula

Use a calculator.

log b = xbx

The solution is about 1.73. Check this in the original equation.

ANSWER

GUIDED PRACTICE for Examples 2 and 3

Solve the equation.

4. 2 = 5x

SOLUTION about 2.32

5. 7 = 159x

SOLUTION about 0.155

Terms and Conceptsthese will be on a quiz

EXAMPLE 4 Solve a logarithmic equation

Solve log (4x – 7) = log (x + 5).5 5

log (4x – 7) = log (x + 5).5 5

4x – 7 = x + 5

3x – 7 = 5

3x = 12

x = 4

Write original equation.

Property of equality for logarithmic equations

Subtract x from each side.

Add 7 to each side.

Divide each side by 3.

The solution is 4.ANSWER

SOLUTION

EXAMPLE 5 Exponentiate each side of an equation

5x – 1 = 64

5x = 65

x = 13

SOLUTION

Write original equation.

Exponentiate each side using base 4.

Add 1 to each side.

Divide each side by 5.

Solve (5x – 1)= 3log4

4log4

(5x – 1) = 4 3

(5x – 1)= (5x – 1)= 3log4

b = xlogbx

The solution is 13.ANSWER

GUIDED PRACTICE for Examples 4, 5 and 6

Solve the equation. Check for extraneous solutions.

7. ln (7x – 4) = ln (2x + 11)

SOLUTION 3

8. log (x – 6) = 52

38

9. log 5x + log (x – 1) = 2

5

SOLUTION

SOLUTION

10. log (x + 12) + log x =344

4SOLUTION