Mrs. McConaughy Honors Algebra 2 1
Properties of Logarithms
During this lesson, you will:
Expand the logarithm of a product, quotient, or power
Simplify (condense) a sum or difference of logarithms
Mrs. McConaughy Honors Algebra 2 2
Part 1:
Expanding Logarithms
Mrs. McConaughy Honors Algebra 2 3
The Product Rule
Let M, N, and b be any positive numbers, such that b ≠ 1.
log b (M ∙ N ) = log b M+ log b N
The logarithm of a product is the sum of the logarithms.
Connection: When we multiply exponents with a common base, we add the exponents.
PROPERTY: The Product Rule (Property)
Mrs. McConaughy Honors Algebra 2 4
Example Expanding a Logarithmic Expression Using
Product Rule
log (4x) = log 4 + log xThe logarithm of a product
is
The sum of the
logarithms.
Use the product rule to expand:
a.log4 ( 7 • 9) = _______________
b. log ( 10x) = ________________
= ________________
log4 ( 7) + log 4(9)
log ( 10) + log (x)
1 + log (x)
Mrs. McConaughy Honors Algebra 2 5
Property: The Quotient Rule (Property)
The Quotient RuleLet M, N, and b be any positive
numbers, such that b ≠ 1.
log b (M / N ) = log b M - log b N
The logarithm of a quotient is the difference of the logarithms.
Connection: When we divide exponents with a common base, we subtract the exponents.
Mrs. McConaughy Honors Algebra 2 6
log (x/2) = log x - log 2
Example Expanding a Logarithmic Expression Using
Quotient Rule
The logarithm of a quotient
is
The difference of the
logarithms.
Use the quotient rule to expand:
a.log7 ( 14 /x) = ______________
b. log ( 100/x) = ______________
= ______________
log7 ( 14) - log 7(x)
log ( 100) - log (x)
2 - log (x)
Mrs. McConaughy Honors Algebra 2 7
PROPERTY: The Power Rule (Property)
The Power Rule
Let M, N, and b be any positive numbers, such that b ≠ 1.
log b Mx = x log b M
When we use the power rule to “pull the exponent to the front,” we say we are _________ the logarithmic expression.expanding
Mrs. McConaughy Honors Algebra 2 8
Example Expanding a Logarithmic Expression Using
Power Rule
Use the power rule to expand:
a.log5 74= _______________
b. log √x = ________________
= ________________
4log5 7
log x 1/2
1/2 log x
Mrs. McConaughy Honors Algebra 2 9
Summary: Properties for Expanding Logarithmic
Expressions
Properties of Logarithms
Let M, N, and b be any positive numbers, such that b ≠ 1.
Product Rule:
Quotient Rule:
Power Rule:
log b (M ∙ N ) = log b M+ log b N
log b (M / N ) = log b M - log b N
log b Mx = x log b M
NOTE: In all cases, M > 0 and N >0.
Mrs. McConaughy Honors Algebra 2 10
Check Point: Expanding Logarithmic Expressions
Use logarithmic properties to expand each expression:
a. logb x2√y b. log6 3√x
36y4
log b x2 + logb y1/2
2log b x + ½ logb y
log 6 x1/3 - log636y4
log 6 x1/3 - (log636 + log6y4)
1/3log 6 x - log636 - 4log6y
2
Mrs. McConaughy Honors Algebra 2 11
Check Point: Expanding Logs
Expand:
log 2 3xy2
log 8 26(xy)2
= log 2 3 + log 2 x + 2log 2 y
= log 8 26 + log 8 x2 + log 8 y2
= 6log 8 2 + 2log 8 x + 2log 8 y
NOTE: You are expanding, not condensing (simplifying) these logs.
Mrs. McConaughy Honors Algebra 2 12
Part 2: Condensing (Simplifying) Logarithms
Mrs. McConaughy Honors Algebra 2 13
Part 2: Condensing (Simplifying) Logarithms
To condense a logarithm, we write the sum or difference of two or more logarithms as single expression.
NOTE: You will be using properties of logarithms to do
so.
Mrs. McConaughy Honors Algebra 2 14
Properties for Condensing Logarithmic Expressions (Working Backwards)
Properties of Logarithms
Let M, N, and b be any positive numbers, such that b ≠ 1.
Product Rule:
Quotient Rule:
Power Rule:
log b M+ log b N = log b (M ∙ N)
log b M - log b N = log b (M /N)
x log b M = log b Mx
Mrs. McConaughy Honors Algebra 2 15
Example Condensing Logarithmic Expressions
Write as a single logarithm: a. log4 2 + log 4 32 =
=a. log (4x - 3) – log x =
log 4 64 3
log (4x – 3)
x
Mrs. McConaughy Honors Algebra 2 16
NOTE: Coefficients of logarithms must be 1 before you condense them using the product and quotient rules.
Write as a single logarithm:
a. ½ log x + 4 log (x-1)
b. 3 log (x + 7) – log x
c. 2 log x + log (x + 1)
= log x ½ + log (x-1)4
= log √x (x-1)4
= log (x + 7)3 – log x= log (x + 7)3
x
= log x2 + log (x + 1)= log x2 (x + 1)
Mrs. McConaughy Honors Algebra 2 17
Check Point: Simplifying (Condensing) Logarithms
a.log 3 20 - log 3 4 =
b. 3 log 2 x + log 2 y =
c. 3log 2 + log 4 – log 16 =
log 3 (20/4) = log 3 5
log 2 x 3y
log 23 + log 4 – log 16 = log 32/16 =log 2
Mrs. McConaughy Honors Algebra 2 18
Example 1 Identifying the Properties of LogarithmsState the property or properties used to rewrite each expression:
Property:____________________________log 2 8 - log 2 4 = log 2 8/4 = log 2 2 = 1
Property:____________________________log b x3 y = log b x3 + log b 7 = 3log b x + log b 7
Property:____________________________
log 5 2 + log 5 6 = log 512
Quotient Rule (Property)
Product Rule/Power Rule
Product Rule (Property)
Sometimes, it is necessary to use more than one property of logs when you expand/condense an expression.
Mrs. McConaughy Honors Algebra 2 19
Example Demonstrating Properties of Logs
Use log 10 2 ≈ 0.031 and log 10 3 ≈ 0.477 to approximate the following:
a. log 10 2/3 b. log 10 6 c. log 10 9 log10 2 – log10 3
0.031 – 0.477
0.031 – 0.477
– 0.466
Mrs. McConaughy Honors Algebra 2 20
Homework Assignment:
Properties of Logs