Reteach 7-5

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600.352. “Bring down” the exponent to multiply. If x y, then log x log y x. 0 and y. 0 . Reteach. Exponential and Logarithmic Equations and Inequalities. 7-5 ...
Name LESSON

7-5

Date

Class

Reteach Exponential and Logarithmic Equations and Inequalities

An exponential equation contains an expression that has a variable as an exponent. 5 x ⫽ 25 is an exponential equation. x ⫽ 2, since 5  2  ⫽ 25. If x ⫽ y, then log x ⫽ log y  x ⬎ 0 and y ⬎ 0 .

Remember: You can take the logarithm of both sides of an exponential equation. Then use other properties of logarithms to solve. Solve 6 x ⫹ 2 ⫽ 500. Step 1 Since the variable is in the exponent, take the log of both sides. 6 x ⫹ 2 ⫽ 500 x⫹2 ⫽ log 500 log 6 Step 2

Use the Power Property of Logarithms: log a p ⫽ p log a. x⫹2 ⫽ log 500 log 6  x ⫹ 2  log 6 ⫽ log 500 “Bring down” the exponent to multiply.

Step 3

Isolate the variable. Divide both sides by log 6.  x ⫹ 2  log 6 ⫽ log 500 log 500 x ⫹ 2 ⫽ _______ log 6 Solve for x. Subtract 2 from both sides. log 500 x ⫽ _______ ⫺ 2 log 6 Use a calculator to approximate x. x  1.468

Step 4

Step 5 Step 6

Use a calculator to check. 6 1.468 ⫹ 2  499.607

Solve and check. 1. 4

⫺x

⫽ 32

log 4

⫺x

⫽ log 32

log 3

⫺x log 4 ⫽ log 32

4x

 ⫺2.5 

⫽ 32

Copyright © by Holt, Rinehart and Winston. All rights reserved.

a207c07-5_rt.indd 38

⫽ log 90

4x log 3 ⫽ log 90

x ⫽ ⫺2.5 4⫺

3. 5 x ⫺ 3 ⫽ 600

2. 3 4x ⫽ 90

x  1.024 34

 1.024 

38

 90.01

log 5 x

x⫺3

⫽ log 600

⫺ 3  log 5 ⫽ log 600

x  6.975 5 6.975 ⫺ 3  600.352

Holt Algebra 2

5/15/06 1:38:35 PM Process Black

Name LESSON

7-5

Date

Class

Reteach Exponential and Logarithmic Equations and Inequalities (continued)

A logarithmic equation contains a logarithmic expression that has a variable. log 5 x ⫽ 2 is a logarithmic equation. 2 x ⫽ 25, since 5 ⫽ 25.

Combine and use properties of logarithms to solve logarithmic equations. Solve: log 80x ⫺ log 4 ⫽ 1 Step 1 Use the Quotient Property of Logarithms. log 80x ⫺ log 4 ⫽ 1 80x ⫽ 1 log ____ 4 Step 2 Simplify. 80x ⫽ 1 log ____ 4 log 20x ⫽ 1 Step 3 Use the definition of the logarithm: x if b ⫽ a, then log b a ⫽ x.

x logx ⫺ logy ⫽ log __ y

Remember: Use 10 as the base when the base is not given.

log 10 20x ⫽ 1 1 10 ⫽ 20x

Step 4 Solve for x. Divide both sides by 20. 10 ⫽ 20x 1⫽x __ 2 Solve and check. 4 4. log 3 x ⫽ 8

5. log 4 ⫹ log  x ⫹ 2  ⫽ 2

4 log 3 x ⫽ 8

log 4  x ⫹ 2  ⫽ 2

8 log 3 x ⫽ __ 4

log 10  4x ⫹ 8  ⫽ 2 4x ⫹ 8 ⫽ 10 2

6. log 75x ⫺ log 3 ⫽ 1

 

75x ⫽ 1 log ____ 3 log 25x ⫽ 1 1

4x ⫹ 8 ⫽ 100

10 ⫽ 25x

32 ⫽ x

4x ⫽ 92

x⫽9

x ⫽ 23

10 ⫽ 25x 2 x ⫽ __ 5

Copyright © by Holt, Rinehart and Winston. All rights reserved.

a207c07-5_rt.indd 39

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