5.4 Exponential and Logarithmic Equations

Essential Questions: How do we solve exponential and logarithmic equations?

Solving Simple Equations

Original Rewritten Solution

Equation Equation

5a. 2 32 2 2 5x x x b. ln ln 3 0 ln ln 3 3x x x

21c. 9 3 3 2

3

xx x

d. 7 ln ln 7 ln 7x xe e x

ln 3 3e. ln 3 xx e e x e 10log 1 1

10

1f. log 1 10 10 10

10xx x

Strategies for Solving Exponential and Logarithmic Equations

1. Rewrite the given equation in a form that allows the use of the One-to-One Properties of exponential or logarithmic functions.

2. Rewrite an exponential equation in logarithmic form and apply the Inverse Property of logarithmic functions.

3. Rewrite a logarithmic equation in exponential form and apply the Inverse Property of exponential functions.

Solving Exponential Equations

Solve each equation and approximate the result to three decimal places.

a. b. 72xe 3(2 ) 42x

Solution:

a. 72xe ln ln 72xe Take natural log of

each side.

ln 72x 4.277x Use a calculator.

b. 3(2 ) 42x

2 14x

2 2log 2 log 14x

2log 14x

3.807x

Solving an Exponential Equation

Solve and approximate the result to three decimal places.

5 60xe

Solution:

5 60xe Original equation.

55xe Subtract 5 from each side.

ln ln 55xe Take natural log of each side.

ln 55x Inverse Property.

4.007x Use a calculator.

Solve and approximate the result to three decimal places.

2 52(3 ) 4 11n

2 52(3 ) 4 11n 2 52(3 ) 15n

2 5 153

2n

2 53 3

15log 3 log

2n

3

152 5 log

2n

32 5 log 7.5n

3

5 1log 7.5

2 2n

3.417n

Solving a Logarithmic Equation

a. Solve ln x = 2 b. Solve 3 3log (5 1) log ( 7)x x

Solution:

a.

ln 2

ln 2x

x

e e

Exponentiate each side.2x e

b. 3 3log (5 1) log ( 7)x x

5 1 7x x 4 8x 2x Check this in the

original equation.

Solving a Logarithmic Equation

Solve and approximate the result to three decimal places.

5 2ln 4x

Solution:

5 2ln 4x 2ln 1x

1ln

2x

ln 1 2xe eExponentiate each side.

1 2x e

0.607x

Solving a Logarithmic Equation

Solve 52log 3 4.x

Solution:

52log 3 4x

5log 3 2x

5log 3 25 5x Exponentiate each side (base 5).

3 25x 25

3x

Checking for Extraneous Solutions

Solve 10 10log 5 log ( 1) 2x x

Solution:

10 10log 5 log ( 1) 2x x

10log [5 ( 1)] 2x x 2

10log (5 5 ) 210 10x x 25 5 100x x

25( 20) 0x x

( 4)( 5) 0x x

4 5x x

The solutions appear to be 5 and -4. However, when you check these in the original equation, only x = 5 works.