Section 5.4 – Properties of Logarithms
Objectives
• Apply the properties of logarithms to expand expressions involving the logarithm of a
product/quotient/power into a sum/difference of logarithms.
• Apply the properties of logarithms to condense a sun/difference of logarithms into a
single logarithm.
• Use the change of base formula to convert a logarithm with any allowable base to a
logarithm with any allowable base. (In particular, convert to base 10 or base e.)
• Use the one-to-one property of logarithms to solve certain logarithmic equations.
Preliminaries
Carefully write the Product, Quotient, and Power Rules for logarithms.
Write the change of base formula
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Warm-up
Evaluate the following, if they are defined, without a calculator:
1
1. log⁡(10,000)
2. log 5 (517 )
3. log1/2(16)
4. log 3 (0)
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Class Notes and Examples
5.4.1 Use properties of logarithms to expand each expression as much as possible.
(A) log 𝑏 (𝑥 2 𝑦 3 ) =
1
(B)
ln (𝑥) =
(C)
log⁡(√5𝑎) =
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(D)
log(10𝑦𝑧1/3 ) =
(E)
log 2 (
(F)
ln (3𝑒 𝑥 ) =
2𝑥
)
𝑦√𝑧
=
𝑥𝑦 3
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5.4.2
Use properties of logarithms to rewrite the following as a single logarithm.
1
(A) ln(6𝑥) + ln(𝑥) − ln(2𝑥) =
2
(B) log(5𝑧) − log(𝑥) − 3 log(3𝑦) + log(𝑡) =
(C) ln(𝑥) − 2 ln(𝑦) − ln(𝑧) =
1
(D) 2 log 2 (𝑥) + log 2 (𝑦) − 4 log 2 (𝑃) − 3 log 2 (𝑄) + log 2 (𝑧) =
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What is the change of base formula, and when would you want to use it?
5.4.3
Use your calculator to approximate the following to four decimal places.
(A) log 3 (6)
(B) log √7(9)
5.4.4
Use change of base formula and properties of logarithms to rewrite the expression
log 2 (𝑥) + log 4 (𝑦) as a single logarithm.
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What does the logarithm property of equality say?
Under what circumstances can we use the logarithm property of equality to solve
equations?
5.4.5
Solve the following equations. Check your solutions.
(A) log(10) = log⁡(𝑥 2 )
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(B) log 2 (5) = log 2 (𝑥) − log 2 (𝑥 + 1)
(C) ln(𝑥) = log 𝑒 2 (13) (Hint: use the change of base formula)
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Section 5.4 Self-Assessment (Answers on page 257)
1.
(Multiple Choice) Write as a sum/difference and/or multiple of logarithms.
log (
(A)
(B)
(C)
(D)
(E)
2.
10𝑥
)
𝑦3
10 + log(𝑥) − log(3𝑦)
10 + log(𝑥) − 3log(𝑦)
10 log(𝑥) − 3log(𝑦)
1 + log(𝑥) − log(3𝑦)
1 + log(𝑥) − 3log(𝑦)
Rewrite the expression as a single logarithm.
1
2 log(𝐴) − log(𝐵) + log(𝐶) − 5 log(𝐷)
3
3.
(Multiple Choice) Write as a sum/difference and/or multiple of logarithms.
3
𝑎⁡√𝑏
ln ( 5 )
𝑐⁡𝑑
1
(B) ln(𝑎) + ln (3 𝑏) − ln(𝑐) − ln(5𝑑)
1
(D) ln(𝑎) + √ln(𝑏) − ln(𝑐) − (ln(𝑑))5
(A) ln(𝑎) + 3 ln(𝑏) − ln(𝑐) − 5 ln(𝑑)
(C) ln(𝑎) + 3 ln(𝑏) − ln(𝑐) + 5 ln(𝑑)
1
3
(E) None of these
4.
Solve the equation.
log 2 (8) = log 2 (3𝑥) − log 2 (𝑥 − 1)
5.
(Multiple Choice) Solve the equation.
log 7 (3) = log 7 (𝑥) + log 7 (𝑥 − 2)
(A) 𝑥 = 3 only
5
2
(D) 𝑥 = only
(B) 𝑥 = −1, 3 only
(C) 𝑥 = −1 only
(E) There are no solutions
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