Intermediate Algebra
Unit 9: Logarithms
Intermediate Algebra
Unit 9: Logarithms
Objectives:
page
introduction to logarithms
2–4
logarithms & logarithmic functions
5–7
GUIDE: properties of logarithmic functions
8
product & quotient property of logarithms
9 – 11
power property of logarithms
12 – 14
simplifying and solving logarithms
15
solving equations with logarithms
16 – 17
review questions
18 – 19
logarithms & word problems
20 – 23
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Intermediate Algebra
Unit 9: Logarithms
Introduction to Logarithms: Exponential Function
inverse → (switch the x and y )
y = bx
Logarithmic Function x = by
Rewriting Equations into Exponential Form and Logarithmic Form:
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Intermediate Algebra
Unit 9: Logarithms
Common Logarithm:
Change of Base Formula:
Use logarithmic and exponential properties to solve for x: (1)
log2 8 = x
(2)
log2
(4)
log x 64 = 2
(5)
log x 125 = 3
(7)
log7 x = 0
(8)
log81 x =
(10) log x = 7
1 =x 4
1 2
(11) log 100 = x
3
(3)
log 4 256 = x
(6)
log x 6 =
(9)
log64 x =
(12) log x
1 3
2 3
1 = −1 10
Intermediate Algebra
Unit 9: Logarithms
Practice Examples: (1) FILL IN THE BLANK: A logarithm is an
.
(2) Given 103 = 1000, rewrite in logarithmic form: log __ 1000 = _____
Write the following equation in exponential form and logarithmic form respectively: (3) 3 4 = 81
−2 (4) 2 =
(5) log 5 125 = 3
(6) log 4
1 4
1 = −3 64
Use a calculator and the change of base formula to evaluate each of the following: (7) log2 8
(8) log3 27
(9) log5 625
(10) log3
4
1 9
Intermediate Algebra
Unit 9: Logarithms
Logarithms & Logarithmic Functions: Do Now: Write each equation in logarithmic form: 1 (2) 7 −2 = (1) 54 = 625 49
Write each equation in exponential form: 1 (3) log3 81 = 4 (4) log36 6 = 2
Evaluate each expression: (5)
log9 3
(6)
log2
1 8
Solve each equation and check your solution(s):
(7)
log9 x =
3 2
(10) logb 9 = 2
(8)
log 1 x = −3 10
(11) log2 (3 x − 5 ) = log2 (x + 7 )
5
(9)
log3 (2x − 1) = 2
( )
(12) log5 (3 x − 1) = log5 2x 2
Intermediate Algebra
Unit 9: Logarithms
Solve each equation and check your solution(s): (13) log6 (2x − 3 ) = log6 (x + 2)
(14) log5 x =
(16) log2 (4 x − 10 ) = log2 (x − 1)
(17) log10 x 2 − 6 = log10 x
1 2
(
(15) logb 121 = 2
)
6
(
)
(18) log7 x 2 + 36 = log7 100
Intermediate Algebra
Unit 9: Logarithms
Practice Examples: Write each equation in logarithmic form: 1 (2) 8 − 2 = (1) 23 = 8 64
Write each equation in exponential form: 1 1 (4) log5 (3) log9 3 = = −2 25 2
Solve each equation and check your solution(s): 1 =x 64
(5)
log5 25 = x
(6)
log10 1000 = x
(7)
(8)
log8 83 = x
(9)
log3 x = 5
(10) log 1 x = 3
log4
4
(11) logx 3 =
1 2
(12) log6 (4 x + 12) = 2
7
(13) log3 (x + 2) = log3 (3 x )
Intermediate Algebra
Unit 9: Logarithms
Properties of Logarithmic Functions If b, M, and N are positive real numbers, b ≠ 1, and p and x are real numbers, then: Definition
Examples
logb 1 = 0
written exponentially: b0 = 1
logb b = 1
written exponentially: b1 = b
logb b x = x
written exponentially: bx = bx
blog b
x
= x , where x > 0
10log 10
logb MN = logb M + logb N
log 1 yz = log 1 y + log 1 z 5
2 = log4 2 − log4 5 5 7 log8 = log8 7 − log8 x x
log2 6 x = x log2 6
logb M = p logb M p
if and only if
5
log4
M = logb M − logb N N
logb M = logb N
=7
log3 9 x = log3 9 + log3 x 5
logb
7
log5 y 4 = 4 log5 y M=N
log6 (3 x − 4) = log6 (5 x + 2) ∴ ( 3 x − 4 ) = ( 5 x + 2) Common Errors:
logb M ≠ logb M − logb N logb N
logb (M + N) ≠ logb M + logb N
(logb M) ≠ p logb M p
logb M − logb N = logb
M N
logb M cannot be simplified logb N
logb M + logb N = logb MN logb (M + N) cannot be simplified
p logb M = logb Mp (logb M)p cannot be simplified
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Intermediate Algebra
Unit 9: Logarithms
Do Now:
Examples:
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Intermediate Algebra
Unit 9: Logarithms
Product & Quotient Property of Logarithms: Solve each equation and check your solution(s): (1)
log3 5 + log3 x = log3 10
(2)
log 4 x + log 4 9 = log 4 27
(3)
log10 16 − log10 2x = log10 2
(4)
log7 24 − log7 (x + 5 ) = log7 8
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Intermediate Algebra
Unit 9: Logarithms
Solve each equation and check your solution(s): (5)
log3 42 − log3 x = log3 7
(7)
log2 (12x − 21) − log2 x 2 − 3 = 2
(
)
(6)
log2 3 x + log2 5 = log2 30
(8)
log2 (x + 2) − log2 (x − 2) = 1
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Intermediate Algebra
Unit 9: Logarithms
Do Now:
Examples:
12
Intermediate Algebra
Unit 9: Logarithms
Power Property of Logarithms: Solve each equation and check your solution(s): (1)
2 log5 x = log5 9
(3)
log2 x =
1 1 log2 16 + log2 49 2 4
(2)
3 log7 4 = 2 log7 x
(4)
2 log10 6 −
13
1 log10 27 = log10 x 3
Intermediate Algebra
Unit 9: Logarithms
Solve each equation and check your solution(s): (5)
log9 4 + 2 log9 5 = log9 x
(6)
3 log8 2 − log8 4 = log8 x
(7)
log2 x + 2 log2 5 = 0
(8)
log5 10 + log5 12 = 3 log5 2 + log5 x
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Intermediate Algebra
Unit 9: Logarithms
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Intermediate Algebra
Unit 9: Logarithms
Solving Equations with Logarithms: Do Now: Solve for x: (1) log5 125 = x
(2)
log8 64 = x
Use logarithms to solve for x and, if necessary, round your answer to four decimal places: (3)
3x = 7
(4)
2x = 5
(5)
14x = 8
(6)
5 x + 1 = 23
(7)
6x = 1.4
(8)
7x – 3 = 5
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Intermediate Algebra
(10) 3x = 80
(11) 9x = 27
(12) 7x = 343
(13) 6x = 127
(14) 12x = 303
(15) 13x = 2839
(16) 2x = 90
(17) 4x = 512
(18) 3x = 5.2
(19) 11x = 153
(20) 10x = 0
(9)
4x = 2
Unit 9: Logarithms
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Intermediate Algebra
Unit 9: Logarithms
Review Questions: *Study your orange textbook exercises and be sure you can answer these questions: ♦ A logarithm is an _______________________. ♦ Logarithms are used to solve for a variable in the __________________________. ♦ Common (decimal) logs are____________________________________________. ♦ Write the procedure for solving log equations here:
Write each equation in exponential form: (1) 3 = log2 8
(2) − 1 = log5 0.2
Write each equation in log form: (3)
49 = 7 2
(4)
125 = 5 3
Solve each equation for x and round your answer to four decimal places: (5)
5 x = 3 .6
(6)
2 x = 1 .5
(7)
4 x = 21
(8)
7 x +2 = 560
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Intermediate Algebra
Unit 9: Logarithms
Use logarithmic properties to solve the following equations and check your solution(s): (9) log27 6 x =
2 3
1 1 log5 64 + log5 216 3 2
(10)
log2 (7 x − 3 ) = log2 (x + 12)
(12)
log7 32 − log7 (2x ) = log7 4
(11)
log5 x =
(13)
4 log2 x + log2 5 = log2 405
(14)
log10 5 + log10 x = 2
(15)
log3 (x + 3 ) + log3 (x − 2) = log3 14
(16)
log2 (x + 1) + log2 (x − 5 ) = 4
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Intermediate Algebra
Unit 9: Logarithms
Logarithms & Word Problems: Do Now: (1)
When solving for an unknown exponent, use
(2)
Solve for x and round your answer to four decimal places:
(3)
A certain car depreciates in value 16% each year. (a) Write an exponential function to model the depreciation of a car that cost $32,500 when purchased new.
(b)
(4)
3.4x = 180.7
Suppose the car was purchased in 2010. What is the first year the car will be worth less than half its original value? (Solve algebraically.)
Alex invested $2500 at a rate of 2.3% in the bank. If the interest is compounded daily, when will Alex’s money double? Round your answer to the nearest year.
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Intermediate Algebra
Unit 9: Logarithms
1.0246x = 4500
(5)
Solve for x and round your answer to four decimal places:
(6)
(a) Write an equation that shows how much money will be in a savings account that pays 2.75% interest compounded monthly when $500 is invested. (Assuming there are no other withdrawals or deposits.)
(b) Using your equation from part (a), find how long, to the nearest year, the initial investment of $500 must be left in this account in order for the account to have a value over $795.
(7)
Lea bought a car for $38,500. It is expected to depreciate at a rate of 12% per year. After how many years, to the nearest year, will it be worth less than $12,000?
(8)
Ms. Pina bought a painting for $5000. It is expected to appreciate in value at a rate of 4% per year. When, to the nearest year, will the painting be worth more than $6300?
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Intermediate Algebra
(9)
Unit 9: Logarithms
Solve for x. Show your work for full credit. Round your answer to the nearest tenth. 2.5 x = 9.88
(10) Michael invested $2000 at 4.5% interest compounded quarterly. At this rate, how long, to the nearest year, will take for Michael’s money to triple, assuming there are no other withdrawals or deposits? (Provide an algebraic solution showing all work.) r A = P1 + n
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nt
Intermediate Algebra
Unit 9: Logarithms
(11) A piece of machinery valued at $ 475,000 depreciates at a fixed rate of 7.2% per year. At this rate, after how many years, to the nearest year, will the value of the equipment be below $225,000? (Provide an algebraic solution showing all work.)
(12) You buy an autographed limited edition U2 CD for $20.00 with the understanding that it will increase in value at a steady rate of 2.75% per year. At this rate, how long, to the nearest year, will it take for the CD to reach a value of $100? (Provide an algebraic solution showing all work.)
CHALLENGE: Solve for x and round your answer to four decimal places: 2 x = 5 x–2
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Intermediate Algebra
Unit 9: Logarithms
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