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
1
Part 1: Expanding
Logarithms
2
PROPERTY: The Product Rule (Property)
Connection: When we multiply like bases with
exponents, we add the exponents.
3
Example Expanding a Logarithmic
Expression Using Product Rule
is
log (4x) = log 4 + log x
The logarithm of
a product
The sum of the
logarithms.
Use the product rule to expand:
log4 ( 7) + log 4(9)
a.log4 ( 7 • 9) = _______________
log ( 10) + log (x)
b. log ( 10x) = ________________
1 + log (x)
= ________________
4
Property: The Quotient
Rule (Property) p.362
The Quotient Rule
Let A, B, and b be any positive
numbers, such that b ≠ 1.
log b (A / B ) = log b A - log b B
The logarithm of a quotient is the
difference of the logarithms.
Connection: When we divide like bases with
exponents, we subtract the exponents.
5
Example Expanding a Logarithmic
Expression Using Quotient Rule
is
log (x/2) = log x - log 2
The logarithm of
a quotient
The difference of
the logarithms.
Use the quotient rule to expand:
log7 ( 14) - log 7(x)
a.log7 ( 14 /x) = ______________
log ( 100) - log (x)
b. log ( 100/x) = ______________
2
log
(x)
= ______________
6
PROPERTY: The Power Rule
(Property) p. 363
The Power Rule
Let A, and b be any positive numbers, such
that b ≠ 1.
log
b
Ap = p log
b
A
When we use the power rule to “pull the
exponent to the front,” we say we are
expanding
_________ the logarithmic expression.
7
Example Expanding a Logarithmic
Expression Using Power Rule
Use the power rule to expand:
4log5 7
a.log5 74= _______________
log x 1/2
b. log √x = ________________
1/2 log x
= ________________
8
Summary: Properties for
Expanding Logarithmic Expressions
Properties of
Logarithms
Product Rule:
Let A, B, and b be any positive
numbers, such that b ≠ 1.
log b (A ∙ B ) = log b A+ log b B
Quotient Rule: log
Power Rule:
b
( A/B ) = log b A - log b B
log b Ap = p log b A
NOTE: In all cases, A > 0 and B > 0.
9
Check Point: Expanding
Logarithmic Expressions
Use logarithmic properties to expand
each expression:
a. logb x2√y
b. log6 3√x
4
36y
log b x2 + logb y1/2
log 6 x1/3 - log636y4
2log b x + ½ logb y
log 6 x1/3 - (log636 + log6y4)
1/3log 6 x - log636 - 4log6y
2
10
NOTE:
YouPoint:
are expanding,
not condensing
Check
Expanding
Logs
(simplifying) these logs.
Expand:
log
log
= log
2
3xy2
8
26(xy)2 = log
= 6log
2
8
8
3 + log
26 + log
2
x + 2log
2
8
x2 + log
2
y
8
2 + 2log
8
x + 2log
y
8
11
y
Part 2: Condensing
(Simplifying) Logarithms
12
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.
13
Properties for Condensing Logarithmic
Expressions (Working Backwards)
Properties of
Logarithms
Product Rule:
Let A, B, and b be any positive
numbers, such that b ≠ 1.
log b A+ log b B = log b (A ∙ B)
Quotient Rule: log
Power Rule:
b
A - log b B = log b (A/B)
p log b A = log b Ap
14
Example Condensing Logarithmic
Expressions
Write as a single logarithm:
a. log4 2 + log 4 32 = log 4 64
= 3
(4x – 3)
log
a. log (4x - 3) – log x =
x
15
NOTE: Coefficients of logarithms must be 1
before you condense them using the product
and quotient rules.
Write as a single
logarithm:
a.
= log x ½ + log (x-1)4
½ log x + 4 log (x-1) = log √x (x-1)4
= log (x + 7)3 – log x
b. 3 log (x + 7) – log x
= log (x + 7)3
x
c. 2 log x + log (x + 1) = log x2 + log (x + 1)
= log x2 (x + 1)
16
Check Point: Simplifying
(Condensing) Logarithms
a.log
3
20 - log
b. 3 log
2
3
4 =log
x + log
y =
2
3
(20/4) = log
log
2
3
5
x 3y
c. 3log 2 + log 4 – log 16 =
log 23 + log 4 – log 16 = log 32/16 =log 2
17
Sometimes,
it
is
necessary
to
use
Example 1 Identifying the
more than one property of logs when
of Logarithms
youProperties
expand/condense
an expression.
State the property or properties
used to rewrite each expression:
Quotient Rule (Property)
Property:____________________________
log 2 8 - log 2 4 = log 2 8/4 = log 2 2 = 1
Product Rule/Power Rule
Property:____________________________
log
b
x3 y =
log
b
x3 + log
b
7 = 3log
b
x + log
b
7
Product Rule (Property)
Property:____________________________
log 5 2 + log 5 6 = log 512
18
Example Demonstrating Properties
of Logs
Use log 10 2 ≈ 0.031 and log
approximate the following:
10
a. log
6
10
2/3
b. log
10
3 ≈ 0.477 to
c. log
10
9
log10 2 – log10 3
0.031 – 0.477
0.031 – 0.477
– 0.466
19