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LOGARITHMS AND THEIR PROPERTIES
Definition of a logarithm: If ᡶ 㐈 0 and ᡔ is a constant 䙦ᡔ 㐅 1䙧, then ᡷ 㐄 log〩 ᡶ if and only if ᡔげ^ 㐄 ᡶ. In the equation ᡷ 㐄 log〩 ᡶ , ᡷ is referred to as the logarithm, ᡔ is the base, and ᡶ is the argument.
The notation log〩 ᡶ is read “the logarithm (or log) base ᡔ of ᡶ.” The definition of a logarithm indicates that a logarithm is an exponent. ᡷ 㐄 log〩 ᡶ is the logarithmic form of ᡔげ^ 㐄 ᡶ ᡔげ^ 㐄 ᡶ is the exponential form of ᡷ 㐄 log〩 ᡶ
Examples of changes between logarithmic and exponential forms:
Write each equation in its exponential form.
a. 2 㐄 log⡵ ᡶ b. 3 㐄 log⡩⡨䙦ᡶ ㎗ 8䙧 c. log⡳ 125 㐄 ᡶ
Solution:
Use the definition ᡷ 㐄 log〩 ᡶ if and only if ᡔげ^ 㐄 ᡶ.
a. b. 3 㐄 log⡩⡨䙦ᡶ ㎗ 8䙧 if and only if 10⡱^ 㐄 䙦ᡶ ㎗ 8䙧.
c. log⡳ 125 㐄 ᡶ if and only if 5け^ 㐄 125.
Write the following in its logarithmic form: ᡶ 㐄 25⡩ ⡰⁄
Solution:
Use ᡶ 㐄 ᡔげ^ if and only if ᡷ 㐄 log〩 ᡶ.
Equality of Exponents Theorem: If ᡔ is positive real number 䙦ᡔ 㐅 1䙧 such that ᡔけ^ 㐄 ᡔげ, then ᡶ 㐄 ᡷ.
Example of Evaluating a Logarithmic Equation:
Evaluate: log⡰ 32 㐄 ᡶ
Solution:
log⡰ 32 㐄 ᡶ if and only if 2 け^ 㐄 32 Since 32 㐄 2⡳, we have 2 け^ 㐄 2⡳ Thus, by Equality of Exponents, ᡶ 㐄 5
2 㐄 log⡵ ᡶ if and only if 7 ⡰^ 㐄 ᡶ
Logarithms are exponents
Base
ᡶ 㐄 25 ⡩^ ⁄⡰^ ᡡᡘ ᡓᡦᡖ ᡧᡦᡤᡷ ᡡᡘ 12 㐄 log⡰⡳ ᡶ
Exponent
Base
PROPERTIES OF LOGARITHMS: If b, a, and c are positive real numbers, ↄ 㐅 ❸, and n is a real number, then:
- Product: log〩䙦ᡓ · ᡕ䙧 㐄 log〩 ᡓ ㎗ log〩 ᡕ
- Quotient: log〩〨〰 㐄 log〩 ᡓ ㎘ log〩 ᡕ
- Power: log〩 ᡓぁ^ 㐄 ᡦ · log〩 ᡓ
- log〩 1 㐄 0
- log〩 ᡔ 㐄 1
- Inverse 1: log〩 ᡔぁ^ 㐄 ᡦ
- Inverse 2: ᡔ⤢⤥⤗㉷^ ぁ^ 㐄 ᡦ, ᡦ 㐈 0
- One-to-One: log〩 ᡓ 㐄 log〩 ᡕ if and only if ᡓ 㐄 ᡕ
- Change of Base: log〩 ᡓ 㐄 ⤢⤥⤗⤢⤥⤗㉸㉸^ 〨〩 㐄 ⤢⤥⤗ 〨⤢⤥⤗ 〩 㐄 ⤢⤤ 〨⤢⤤ 〩
Examples – Rewriting Logarithmic Expressions Using Logarithmic Properties: Use the properties of logarithms to rewrite each expression as a single logarithm: a. 2 log〩 ᡶ ㎗ ⡩⡰ log〩䙦ᡶ ㎗ 4䙧^ b.^ 4 log〩䙦ᡶ ㎗ 2䙧 ㎘ 3 log〩䙦ᡶ ㎘ 5䙧 Solution: a. 2 log〩 ᡶ ㎗ ⡩⡰ log〩䙦ᡶ ㎗ 4䙧 㐄 log〩 ᡶ⡰^ ㎗ log〩䙦ᡶ ㎗ 4䙧⡩ ⡰⁄^ Power Property 㐄 log〩 㐧ᡶ⡰䙦ᡶ ㎗ 4䙧⡩ ⡰⁄^ 㐱 Product Property
b. 4 log〩䙦ᡶ ㎗ 2䙧 ㎘ 3 log〩䙦ᡶ ㎘ 5䙧 㐄 log〩 䙦ᡶ ㎗ 2䙧⡲^ ㎘ log〩䙦ᡶ ㎘ 5䙧⡱^ Power Property 㐄 log〩䙦け⡸⡰䙧 䙦け⡹⡳䙧ㄠㄙ Quotient Property
Use the properties of logarithms to express the following logarithms in terms of logarithms of ᡶ, ᡷ, and ᡸ.
a. log〩䙦ᡶᡷ⡰䙧 (^) b. log〩けㄘ こ√げㄡ
Solution: a. log〩䙦ᡶᡷ⡰䙧 㐄 log〩 ᡶ ㎗ log〩 ᡷ⡰^ Product Property 㐄 log〩 ᡶ ㎗ 2 log〩 ᡷ Power Property
b. log〩^ ᡶ⡰ ᡸ㒓ᡷ⡳ 㐄 log〩 㐵ᡶ⡰㒓ᡷ㐹 ㎘ log〩 ᡸ⡳^ Quotient Property 㐄 log〩 㐵ᡶ⡰㒓ᡷ㐹 ㎘ log〩 ᡸ⡳^ Quotient Property 㐄 log〩 ᡶ^2 ㎗ log〩 㒓ᡷ ㎘ log〩 ᡸ^5 Product Property 㐄 2 log〩 ᡶ ㎗ ㄗㄘ log〩 ᡷ ㎘ 5 log〩 ᡸ Power Property Other Logarithmic Definitions:
- Definition of Common Logarithm: Logarithms with a base of 10 are called common logarithms. It is customary to write log⡩⡨ ᡶ as log ᡶ.
- Definition of Natural Logarithm: Logarithms with the base of ᡗ are called natural logarithms. It is customary to write log〲 ᡶ as ln ᡶ.
Evaluate the following logarithms for the given values of ᡶ:
- ᡘ䙦ᡶ䙧 㐄 log⡱ ᡶ a. ᡶ 㐄 1 b. ᡶ 㐄 27 c. ᡶ 㐄 0.
- ᡙ䙦ᡶ䙧 㐄 log ᡶ a. ᡶ 㐄 0.01 b. ᡶ 㐄 0.1 c. ᡶ 㐄 30
- ᡘ䙦ᡶ䙧 㐄 ln ᡶ a. ᡶ 㐄 ᡗ (^) b. ᡶ 㐄 ⡩⡱ c. ᡶ 㐄 10
- ᡠ䙦ᡶ䙧 㐄 ln ᡶ a. ᡶ 㐄 ᡗ⡰^ b. ᡶ 㐄 ⡳⡲ c. ᡶ 㐄 1200
- ᡙ䙦ᡶ䙧 㐄 ln ᡗ⡱け a. ᡶ 㐄 ㎘2 b. ᡶ 㐄 0 c. ᡶ 㐄 7.
- ᡘ䙦ᡶ䙧 㐄 log⡰ √ᡶ a. ᡶ 㐄 4 b. ᡶ 㐄 64 c. ᡶ 㐄 5. Use the change of base formula to evaluate the following logarithms: (Round to 3 decimal places.)
- log⡲ 9 54. log⡩ ⡰⁄ 5 55. log⡩⡰ 200 56. log⡱ 0.
Approximate the following logarithms given that log⡳ 2 㐆 0.43068 and log⡳ 3 㐆 0.68261:
- log⡳ 18
- log⡳ √
- log⡳⡩⡰
- log⡳⡰⡱
- log⡳䙦12䙧⡰ ⡱⁄
- log⡳䙦5⡰^ · 6䙧 Use the properties of logarithms to expand the expression:
- log⡲ 6ᡶ⡲
- log 2ᡶ⡹⡱
- log⡳ √ᡶ ㎗ 2
- ln 㒕^ ㄙけ⡳
- ln け⡸⡰け⡹⡰
- ln ᡶ䙦ᡶ ㎘ 3䙧⡰
- ln㐧√2ᡶ䙦ᡶ ㎗ 3䙧⡳㐱
- log⡱〨
ㄘ√〩 〰〱ㄡ
Use the properties of logarithms to condense the expression:
- ㎘ ⡰⡱ ln 3ᡷ
- 5 log⡰ ᡷ
- log⡶ 16ᡶ ㎗ log⡶ 2ᡶ⡰
- log⡲ 6ᡶ ㎘ log⡲ 10
- ㎘2䙦ln 2ᡶ ㎘ ln 3䙧
- 4䙦1 ㎗ ln ᡶ ㎗ ln ᡶ䙧
- 4䙰log⡰ ᡣ ㎘ log⡰䙦ᡣ ㎘ ᡲ䙧䙱
- ⡩⡱ 䙦log⡶ ᡓ ㎗ 2 log⡶ ᡔ䙧
- 3 ln ᡶ ㎗ 4 ln ᡷ ㎗ ln ᡸ
- ln䙦ᡶ ㎗ 4䙧 ㎘ 3 ln ᡶ ㎘ ln ᡷ
True or False? Use the properties of logarithms to determine whether the equation is true or false. If false, state why or give an example to show that it is false.
- log⡰ 4ᡶ 㐄 2 log⡰ ᡶ
- (^) ⤢⤤ ⡩⡨け⤢⤤ ⡳け 㐄 ln ⡩⡰
- log 10⡰け^ 㐄 2ᡶ
- ᡗ⤢⤤ ぇ^ 㐄 ᡲ
- log⡲⡩⡴け 㐄 2 ㎘ log⡲ ᡶ
- 6 ln ᡶ ㎗ 6 ln ᡷ 㐄 ln䙦ᡶᡷ䙧⡴
Practice Problems Answers
Note: Remember that all variables that represent an argument of a logarithm must be greater than 0.
- 3
- 0
- log⡰ 16 㐄 4
- log⡴⡲ 8 㐄 ⡩⡰
- ln 54.60 㐄 4
- 2
- ㎘
- ㎘ 2
- 565
- ㎘ 1
- 380
- 1 ㎗ log ᡶ
- ln ᡶ ㎗ ln ᡷ ㎘ ln ᡸ
- 4 log〩 ᡶ ㎘ 2 log〩 ᡸ
- 1 ㎗ 2 log⡲ ᡶ
- ⡩⡰ log⡱䙦ᡶ ㎘ 2䙧
- 䙦5 ln ᡶ ㎗ 2 ln ᡸ䙧 ㎘ 3 ln ᡷ
- ⡩⡰ 䙦ln 3 ㎗ ln ᡶ䙧 ㎘ ln 7
- log ⡵け
- ln けㄙ こげㄠㄘ
- ln ᡶ⡱
- log⡰⡳け⡱
- log⡲ 4ᡶ⡱
- ln 64ᡶ⡳
- log⡳√⡵ けㄘ
- ㎘1.
- log⡲ 64 㐄 3
- log⡰⡳ 125 㐄 ⡱⡰
- ᡗ⡩^ 㐄 ᡗ
- 3 ⡹⡰^ 㐄 ⡩⡷
- 3
- ⡩⡰
- ㎘
- ㎘ 2
- 7
- ㎘
- 0
- ㎘
- a. 0 b. 3 c. ㎘0.
- a. ㎘2 b. ㎘1 c. 1.
- a. 1 b. ㎘1.099 c. 2.
- a. 2 b. 0.223 c. 7.
- a. ㎘6 b. 0 c. 22.
- a. 1 b. 3 c. 1.
- ㎘2.
- 132
- ㎘1.
- 7959
- 556645
- ㎘0.
- ㎘0.
- 02931
- 11329
- log⡲ 6 ㎗ 4 log⡲ ᡶ
- log 2 ㎘ 3 log ᡶ
- ⡩⡰ log⡳䙦ᡶ ㎗ 2䙧
- ⡩⡱ 䙦ln ᡶ ㎘ ln 5䙧
- ln䙦ᡶ ㎗ 2䙧 ㎘ ln䙦ᡶ ㎘ 2䙧
- ln ᡶ ㎗ 2 ln䙦ᡶ ㎘ 3䙧
- ⡩⡰ 䙦ln 2 ㎗ ln ᡶ䙧 ㎗ 5 ln䙦ᡶ ㎗ 3䙧
- 2 log⡱ ᡓ ㎗ ⡩⡰ log⡱ ᡔ ㎘ log⡱ ᡕ ㎘ 5 log⡱ ᡖ
- ln 䙲 (^) ⡱げ⡩䙳⡰ ⡱
⁄
- log⡰ ᡷ⡳
- log⡶ 32ᡶ⡱
- log⡲⡱け⡳
- ln (^) ⡲け⡷ㄘ
- 4 ㎗ ln ᡶ⡶
- log⡰ 䙲 (^) 〸⡹ぇ〸䙳⡲
- log⡶ √ᡓᡔㄙ ⡰
- ln䙦ᡶ⡱ᡷ⡲ᡸ䙧
- ln け⡸⡲けㄙげ
- False. log⡰ 4ᡶ 㐄 2 ㎗ log⡰ ᡶ
- False. ln (^) ⡩⡨け⡳け 㐄 ln ⡩⡰
- True
- True
- True
- True
