Holt McDougal Algebra 2 7-5 Exponential and Logarithmic Equations and Inequalities 7-5 Exponential and Logarithmic Equations and Inequalities Holt Algebra

Holt McDougal Algebra 2

7-5 Exponential and Logarithmic Equations and Inequalities7-5 Exponential and Logarithmic Equations and Inequalities

Holt Algebra 2

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Lesson QuizLesson Quiz



7-5 Exponential and Logarithmic Equations and Inequalities

Opener-SAME SHEET-12/12

Solve for x

1.4x – 12 = 2x + 14

2. 3(x – 6) = 8 + 5x



Example 2: Subtracting Logarithms

log749 – log77

log

27 + log13

13

19

c. log2 ( )5 1

2

c. 5log510

Evaluate log816.



7-4 Hmwk Quiz1. log

749 – log77

2. Evaluate log32

8.

3. log5252

4. log64 + log

69



Solve exponential and logarithmic equations and equalities.

Solve problems involving exponential and logarithmic equations.

Objectives



• 7-5 Explore



exponential equationlogarithmic equation

Vocabulary



Review Properties



An exponential equation is an equation containing one or more expressions that have a variable as an exponent. To solve exponential equations:

• Try writing them so that the bases are all the same.

• Take the logarithm of both sides.



When you use a rounded number in a check, the result will not be exact, but it should be reasonable.

Helpful Hint



Solve and check.98 – x = 27x – 3

Example 1A: Solving Exponential Equations

x = 5

98 – x = 27x – 3



Solve and check.4x – 1 = 5

Example 1B: Solving Exponential Equations

Check Use a calculator.

The solution is x ≈ 2.161.

x = 1 + ≈ 2.161log5log4



Solve and check.

32x = 27

Check It Out! Example 1a

x = 1.5

7–x = 21

x = – ≈ –1.565log21log7

23x = 15

x ≈ 1.302



Card Problems



Suppose a bacteria culture doubles in size every hour. How many hours will it take for the number of bacteria to exceed 1,000,000?

Example 2: Biology Application

Solve 2n > 106

At hour 0, there is one bacterium, or 20 bacteria. At hour one, there are two bacteria, or 21 bacteria, and so on. So, at hour n there will be 2n bacteria.

Write 1,000,000 in scientific annotation.

Take the log of both sides.log 2n > log 106



Example 2 Continued

Use the Power of Logarithms.

log 106 is 6.nlog 2 > 6

nlog 2 > log 106

6log 2n > Divide both sides by log 2.

60.301n > Evaluate by using a calculator.

n > ≈ 19.94 Round up to the next whole number.

It will take about 20 hours for the number of bacteria to exceed 1,000,000.



You receive one penny on the first day, and then triple that (3 cents) on the second day, and so on for a month. On what day would you receive a least a million dollars.

Solve 3n – 1 > 1 x 108

$1,000,000 is 100,000,000 cents. On day 1, you would receive 1 cent or 30 cents. On day 2, you would receive 3 cents or 31 cents, and so on. So, on day n you would receive 3n–1 cents.

Write 100,000,000 in scientific annotation.

Take the log of both sides.log 3n – 1 > log 108

Check It Out! Example 2



Use the Power of Logarithms.

log 108 is 8.(n – 1)log 3 > 8

(n – 1) log 3 > log 108

8log 3n – 1 > Divide both sides by log 3.

Evaluate by using a calculator.

n > ≈ 17.8 Round up to the next whole number.

Beginning on day 18, you would receive more than a million dollars.

Check It Out! Example 2 Continued

8log3n > + 1



A logarithmic equation is an equation with a logarithmic expression that contains a variable. You can solve logarithmic equations by using the properties of logarithms.

Raise to Same base



Review the properties of logarithms from Lesson 7-4.

Remember!



Solve.

Example 3A: Solving Logarithmic Equations

log6(2x – 1) = –1

7 12

x =

Solve.log

4100 – log4(x + 1) = 1

x = 24



Opener-SAME SHEET-12/13Express each as a single logarithm.

1. log69 + log

624 log

6216 = 3

2. log3108 – log

34

Simplify.

3. log2810,000

log327 = 3

30,000

4. log44x –1 x – 1

5. 10log125 125



Solve.

Example 3C: Solving Logarithmic Equations

log5x 4 = 8

x = 25



Solve.Example 3D: Solving Logarithmic Equations

log12

x + log12

(x + 1) = 1

x(x + 1) = 12

log12

x + log12

(x +1) = 1x = 3 or x = –4

log12

x + log12

(x +1) = 1

log12

3 + log12

(3 + 1) 1log

123 + log

124 1

log12

12 1

The solution is x = 3.1 1

log12

( –4) + log12

(–4 +1) 1

log12

( –4) is undefined.

x



Solve.

3 = log 8 + 3log x

Check It Out! Example 3a

5 = x

2log x – log 4 = 0

x = 2



Watch out for calculated solutions that are not solutions of the original equation.

Caution



Use a table and graph to solve 2x + 1 > 8192x.

Example 4A: Using Tables and Graphs to Solve Exponential and Logarithmic Equations and Inequalities

Use a graphing calculator. Enter 2^(x + 1) as Y1 and 8192x as Y2.

In the table, find the x-values where Y1 is greater than Y2.

In the graph, find the x-value at the point of intersection.

The solution set is {x | x > 16}.



In the table, find the x-values where Y1 is equal to Y2.


Check It Out! Example 4aUse a table and graph to solve 2x = 4x – 1.

Use a graphing calculator. Enter 2x as Y1 and 4(x – 1) as Y2.

The solution is x = 2.



Team Problems

Solve.

1. 43x–1 = 8x+1

2. 32x–1 = 20

3. log7(5x + 3) = 3

4. log(3x + 1) – log 4 = 2

5. log4(x – 1) + log

4(3x – 1) = 2

x ≈ 1.86

x = 68

x = 133

x = 3

x = 5 3



Opener-SAME SHEET-1/26In 2000, the world population was 6.08 billion and was increasing at a rate 1% each year.1. Write a function for world population. Does

the function represent growth or decay?P(t) = 6.08(1.01)t

2. Use a table to predict the population in 2020.

≈ 7.41 billionThe value of a $3000 computer decreases about 30% each year.3. Write a function for the computer’s value.

Does the function represent growth or decay?

4. Use a graph to predict the value in 4 years.

V(t)≈ 3000(0.7)t ≈ $720.30



Review

• 9x = 3x-2 4x = 10 log6(2x + 3) = 3



log(x + 70) = 2log( )

In the table, find the x-values where Y1 equals Y2.


x 3

Use a graphing calculator. Enter log(x + 70) as Y1 and 2log( ) as Y2. x

3

The solution is x = 30.

Example 4B



In the table, find the x-values where Y1 is greater than Y2.


Check It Out! Example 4bUse a table and graph to solve 2x > 4x – 1.

Use a graphing calculator. Enter 2x as Y1 and 4(x – 1) as Y2.

The solution is x < 2.



Lesson Quiz: Part II

6. A single cell divides every 5 minutes. How long will it take for one cell to become more than 10,000 cells?

7. Use a table and graph to solve the equation 23x = 33x–1.

70 min

x ≈ 0.903

Documents

Holt McDougal Algebra 2 7-5 Exponential and Logarithmic Equations and Inequalities 7-5 Exponential and Logarithmic Equations and Inequalities Holt Algebra