Algebra II 4 5 Study Guide Exponential and Logarithmic ... II 4_5...Algebra II 4-5 Study Guide Page !1 of 13! Exponential and Logarithmic Equations and Inequalities Attendance Problems

Algebra II 4-5 Study Guide Page ! of !1 13

Exponential and Logarithmic Equations and Inequalities !Attendance Problems. Solve the following.

1. ! 2. ! 3. log 10,000 = x

!!!!!!!!• I can solve exponential and logarithmic equations and equalities. • I can solve problems involving exponential and logarithmic equations. !

Common Core CCSS.MATH.CONTENT.HSF.LE.A.4 For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. CCSS.MATH.CONTENT.HSA.CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. !

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

log16 x =32

logx1.331= 3

Vocabulary

exponential equation logarithmic equation

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

•  Take the logarithm of both sides.


Video Example 1. Solve and Check. A. ! B. ! !!!!!!

27x = 3x+8 7x−3 = 350

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

Helpful Hint


!

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. If b x = b y , then x = y (b ≠ 0, b ≠ 1) .

• Take the logarithm of both sides. If a = b, then log a = log b (a > 0, b > 0) .

1E X A M P L E Solving Exponential Equations

Solve and check.

A 8 x = 2 x + 6 ( 2 3 )

x = 2 x + 6 Rewrite each side with the same base; 8 is a power of 2.

2 3x = 2 x + 6 To raise a power to a power, multiply exponents.

3x = x + 6 Bases are the same, so the exponents must be equal.

x = 3 Solve for x.

Check −−−−−−− 8 x 2 x + 6 8 3 2 3 + 6 8 3 2 9 512 512 ✔ The solution is x = 3.

B 5 x - 2 = 200 log 5 x - 2 = log 200

200 is not a power of 5, so take the log of both sides.

(x - 2) log 5 = log 200 Apply the Power Property of Logarithms.

x - 2 = log 200

_ log 5

Divide both sides by log 5.

x = 2 + log 200

_ log 5

≈ 5.292

Check Use a calculator.

The solution is x ≈ 5.292.

Solve and check. 1a. 3 2x = 27 1b. 7 -x = 21 1c. 2 3x = 15

Exponential and Logarithmic Equations and Inequalities

ObjectivesSolve exponential and logarithmic equations and inequalities.

Solve problems involving exponential and logarithmic equations.

Vocabularyexponential equationlogarithmic equation

Who uses this?Exponential scales are used to measure light in photography. (See Exercise 40.)

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

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266 Chapter 4 Exponential and Logarithmic Functions

4-5CC.9-12.F.LE.4 For exponential models, express as a logarithm the solution to abct = d … evaluate the logarithm using technology. Also CC.9-12.A.CED.1, CC.9-12.A.REI.11

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Example 1. Solve and Check. A. ! B. ! !!!!!!!!!!!!!!!!!

Guided Practice. Solve and Check. 4. ! 5. ! 6. ! !!!!!!!!!!!!!

98−x = 27x−3 4 x−1 = 5

32x = 27 7− x = 21 23x = 15


Video Example 2. You can choose a prize of either a $25,000 car or one penny on the first day, triple that (3 cents) on the 2nd day, and sol for a month. On what day would you receive more than the value of the car? !!!!!!!!!!!!


2E X A M P L E Money Application

You can choose a prize of either a $20,000 car or one penny on the first day, double that (2 cents) on the second day, and so on for a month. On what day would you receive more than the value of the car?

$20,000 is 2,000,000 cents. On day 1, you would receive 1 cent, or 2 0 cents. On day 2, you would receive 2 cents, or 2 1 cents, and so on. So, on day n you would receive 2 n-1 cents.

Solve 2 n - 1 > 2 × 10 6 . Write 2,000,000 in scientific notation.

log 2 n - 1 > log (2 × 10 6 ) Take the log of both sides.

(n - 1) log 2 > log 2 + log 10 6 Use the Power Property and Product Property.

(n - 1) log 2 > log 2 + 6 log 10 6 is 6.

n - 1 > log 2 + 6

_ log 2


n > ≈ 0.301 + 6 _ 0.301

+ 1 Evaluate by using a calculator.

n > ≈ 21.93 Round this up to the next whole number.

Beginning on day 22, you would receive more than the value of the car.

Check On day 22, you would receive 2 22 - 1 cents.

2 22 - 1 = 2 21 = 2,097,152 cents, or $20,971.52.

2. In Example 2, suppose that you receive triple the amount each day. On what day would you receive at least a million dollars?

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.

If log b x = log b y then x = y

3E X A M P L E Solving Logarithmic Equations

Solve.

A log 3 (x - 5) = 2

3 log 3 (x - 5) = 3 2 Use 3 as the base for both sides.

x - 5 = 9 Use inverse properties to remove 3 to the log base 3.

x = 14 Simplify.

B log 45x - log 3 = 1

log ( 45x _ 3

) = 1 Write as a quotient.

log (15x) = 1 Divide.

10 log 15x = 10 1 Use 10 as a base for both sides.

15x = 10 Use inverse properties on the left side.

x = 2 _ 3

Review the properties of logarithms from the previous lesson.

4-5 Exponential and Logarithmic Equations and Inequalities 267

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Example 2. 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? !!!!!!!!!!!!7. Guided Practice. 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. !!!!!!!!!!!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.

!


Video Example 3. Solve each equation. A. ! B. log 30x - log 6 = 4 !!!!!!!!!!!!

C. ! D. log x + log (x + 3) = 1 !!!!!!!!!!!

log4 x −1( ) = 3

log3 x2 = 8


2E X A M P L E Money Application

You can choose a prize of either a $20,000 car or one penny on the first day, double that (2 cents) on the second day, and so on for a month. On what day would you receive more than the value of the car?

$20,000 is 2,000,000 cents. On day 1, you would receive 1 cent, or 2 0 cents. On day 2, you would receive 2 cents, or 2 1 cents, and so on. So, on day n you would receive 2 n-1 cents.

Solve 2 n - 1 > 2 × 10 6 . Write 2,000,000 in scientific notation.

log 2 n - 1 > log (2 × 10 6 ) Take the log of both sides.

(n - 1) log 2 > log 2 + log 10 6 Use the Power Property and Product Property.

(n - 1) log 2 > log 2 + 6 log 10 6 is 6.

n - 1 > log 2 + 6

_ log 2


n > ≈ 0.301 + 6 _ 0.301

+ 1 Evaluate by using a calculator.

n > ≈ 21.93 Round this up to the next whole number.

Beginning on day 22, you would receive more than the value of the car.

Check On day 22, you would receive 2 22 - 1 cents.

2 22 - 1 = 2 21 = 2,097,152 cents, or $20,971.52.

2. In Example 2, suppose that you receive triple the amount each day. On what day would you receive at least a million dollars?

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.

If log b x = log b y then x = y

3E X A M P L E Solving Logarithmic Equations

Solve.

A log 3 (x - 5) = 2

3 log 3 (x - 5) = 3 2 Use 3 as the base for both sides.

x - 5 = 9 Use inverse properties to remove 3 to the log base 3.

x = 14 Simplify.

B log 45x - log 3 = 1

log ( 45x _ 3

) = 1 Write as a quotient.

log (15x) = 1 Divide.

10 log 15x = 10 1 Use 10 as a base for both sides.

15x = 10 Use inverse properties on the left side.

x = 2 _ 3

Review the properties of logarithms from the previous lesson.



Solve.

C log 4 x 2 = 7

2 log 4 x = 7 Power Property of Logarithms

log 4 x = 7 _ 2

Divide both sides by 2 to isolate lo g 4 x.

x = 4 7 _ 2

Definition of a logarithm

x = ( 2 2 ) 7 _ 2

4 is a power of 2.

x = 2 7 , or 128

D log x + log (x + 9) = 1

log x (x + 9) = 1 Product Property of Logarithms.

10 log x(x + 9) = 10 1 Exponential form

x (x + 9) = 10 Use the inverse properties.

x 2 + 9x - 10 = 0 Multiply and collect terms.

(x - 1) (x + 10) = 0 Factor.

x - 1 = 0 or x + 10 = 0 Set each of the factors equal to zero.

x = 1 or x = -10 Solve.

Check Check both solutions in the original equation. −−−−−−−−−−−−−−

log x + log (x + 9) 1 −−−−−−−−−−−−−−−−−−−

log x + log (x + 9) 1X

log 1 + log (1 + 9) 1 log (-10) + log (-10 + 9) 1 ✘ log 1 + log 10 1 ! log(-10) is undefined.

0 + 1 1 1 1 ✔The solution is x = 1.

3a. Solve 3 = log 8 + 3 log x. 3b. Solve 2 log x - log 4 = 0.

4E X A M P L E Using Tables and Graphs to Solve Exponential and Logarithmic Equations and Inequalities

Use a table and graph to solve.

A 2 2x = 1024Use a graphing calculator. Enter 2^(2X) as Y1 and 1024 as Y2.

2000

10

0

0

In the table, find the x-value where Y1 and Y2 are equal.

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

The solution is x = 5.

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




Example 3. Solve each equation. A. ! B. ! !!!!!!!!!!!!C. ! D. ! !!!!!!!!!!!!!

log6(2x −1) = −1 log4 100 − log4 (x +1) = 1

log5 x4 = 8 log12 x + log12 x +1( ) = 1


Guided Practice. Solve each equation. 8. 3 = log 8 + 3 log x 9. 2 log x - log 4 = 0 !!!!!!!!!!!!!

!Video Example 4. Use tables and graphs to solve.

A. ! B. log x - log 5 > log 35 !!!

32x = 6561


Caution


Solve.

C log 4 x 2 = 7

2 log 4 x = 7 Power Property of Logarithms

log 4 x = 7 _ 2

Divide both sides by 2 to isolate lo g 4 x.

x = 4 7 _ 2

Definition of a logarithm

x = ( 2 2 ) 7 _ 2

4 is a power of 2.

x = 2 7 , or 128

D log x + log (x + 9) = 1

log x (x + 9) = 1 Product Property of Logarithms.

10 log x(x + 9) = 10 1 Exponential form

x (x + 9) = 10 Use the inverse properties.

x 2 + 9x - 10 = 0 Multiply and collect terms.

(x - 1) (x + 10) = 0 Factor.

x - 1 = 0 or x + 10 = 0 Set each of the factors equal to zero.

x = 1 or x = -10 Solve.

Check Check both solutions in the original equation. −−−−−−−−−−−−−−

log x + log (x + 9) 1 −−−−−−−−−−−−−−−−−−−

log x + log (x + 9) 1X

log 1 + log (1 + 9) 1 log (-10) + log (-10 + 9) 1 ✘ log 1 + log 10 1 ! log(-10) is undefined.

0 + 1 1 1 1 ✔The solution is x = 1.

3a. Solve 3 = log 8 + 3 log x. 3b. Solve 2 log x - log 4 = 0.

4E X A M P L E Using Tables and Graphs to Solve Exponential and Logarithmic Equations and Inequalities


A 2 2x = 1024Use a graphing calculator. Enter 2^(2X) as Y1 and 1024 as Y2.

2000

10

0

0

In the table, find the x-value where Y1 and Y2 are equal.


The solution is x = 5.





B log x - log 2 ≤ log 75Use a graphing calculator. Enter log x - log 2 as Y1 and log 75 as Y2.

3

3000

0

Y2 = log 75

Y1 = log x - log 2

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


The solution set is ⎧

⎨

⎩ x | 0 < x ≤ 150

⎫

⎬

⎭ .

Check Use algebra.

log x - log 2 ≤ log 75

log ( x _ 2

) ≤ log 75 Quotient Property of Logarithms

10 log ( x _

2 ) ≤ 10 log 75 Use 10 as a base for both sides.

x _ 2

≤ 75 Inverse Property

x ≤ 150 ✔ log x is only defined for x > 0.


4a. 2 x = 4 x - 1 4b. 2 x > 4 x - 1 4c. log x 2 = 6

THINK AND DISCUSS 1. Explain why a and b must be equal if log a = log b (a > 0, b > 0) .

2. Give only the first step you would use to solve each equation.

a. log x 5 = 10 b. log 2x + log 2 = 1

c. x 4 = 100 d. log (x + 1000) = 2

e. log (x + 4) + log x = 2 f. log 6 (x + 6) = 3

3. Explain whether a logarithmic equation can have a negative number as a solution. Justify your answer. Give an example, if possible.

4. GET ORGANIZED Copy and complete the graphic organizer. Write the strategies and points to remember in your own words for both exponential and logarithmic equations.

Strategies tosolve

Points toremember

Strategies tosolve

Points toremember

Equation

Exponential Logarithmic


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Example 4. Use tables and graphs to solve.

A. ! B. !

!Guided Practice. Use tables and graphs to solve.

10. ! 11. ! 12. ! !!!!4-5 Assignment (p 270) 21-26, 28-33, 48-50.

2x+1 > 8192x log x + 70( ) = 2 log x3

⎛⎝⎜

⎞⎠⎟

2x = 4 x−1 2x > 4 x−1 log x2 = 6

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Algebra II 4 5 Study Guide Exponential and Logarithmic ... II 4_5...Algebra II 4-5 Study Guide Page !1 of 13! Exponential and Logarithmic Equations and Inequalities Attendance Problems