Lesson 5.4: Exponential & Logarithmic Equations 2 Basic - - PowerPoint PPT Presentation

Lesson 5.4: Exponential & Logarithmic Equations

2 Basic strategies for solving Exponential and Logarithmic equations: 1.Use 1-to-1 properties:

• 2. Use Inverse properties: a

x a x

a x

a x log

log  

a a iff x y x y iff x y

x y a a

    log log

Ex 1: Solve for x.

Using 1-1 properties

a.2x = 32

• b. ln x – ln 3 = 0

c.

2 25

x 

x  5 ln ln x  3 x  3

1 3 9

F H I K 

x

1 3 9

x 

3 32

  x

  x 2 x  2

Ex 1: Solve for x. Using Inverse properties

• d. ex = 7
• e. ln x = -3
• f. log10 x = -1

ln ln ex  7 x e ln ln  7 x  ln7 1946 .

e e

x ln  3

x e 

3 .050

10 10

10

1 log x  

x 



10 1  1

10

Recap of Strategies Used: 1.Rewrite original equation in a form that allows the use of 1-to-1properties. 2.Rewrite an exponential equation in logarithmic form and apply Inverse properties. 3.Rewrite a logarithmic equation in exponential form and apply Inverse properties.

Ex 2: Solve for x.

Exponential Equations

• a. 4x = 72
• b. 3(2)x = 42

log log

4 4

4 72

x 

x  log log

10 10

72 4

 3085 .

3 3 2 14

x 

log log

2 2

2 14

x 

x  log log

10 10

14 2

 3807 .

• c. ex + 5 = 60

Ex 2: Solve for x.

ex  55

ln ln ex  55 x  ln55

 4 007 .

Ex 2: Solve for x. Logarithmic Equations

• d. ln x = 2
• e. log3 (5x – 1) = log3 (x + 7)

e e

x ln  2

x e 

2

 7389 .

log log

a a

x y iff x y  

5 1 7 x x   

1 1

5 8 x x  

x x

4 8 x  x  2

Ex 2: Solve for x.

• f. 5 + 2 ln x = 4

Homework: p.403 #9-20 all

5 5

2 1 ln x  

2 2

ln x   1 2 e e

x ln  1 2

x e 

1 2

 0 607 .