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Section 5.4 Properties of Logarithmic Functions 1
Copyright © 2016 Pearson Education, Inc.
Section 5.4 – Properties of Logarithmic Functions This section covers some properties of logarithmic function that are very similar to the rules for exponents. Properties of Logarithms For any positive number M and N, and any logarithmic base a, Product Rule: ( )log log loga a aM N M N⋅ = +
Quotient Rule: log log loga a aM M NN
⎛ ⎞ = −⎜ ⎟⎝ ⎠
Product Rule: log logpa aM p M= ⋅
Example 1: Express as a sum of logarithms by using the Product Rule.
( )3
3
9 27 (By the Product Rule)
(By the definition o
lo
g )
g
f lo
⋅ =
=
Example 2: Express as a single logarithm.
32 2 log (By the Produlog ct Rule)p q+ =
Example 3: Express 3log 11a
− as a product. Compare this to the left side of the Power Rule: log logp
a aM p M= ⋅ .
M = and p =
Now inserting these in to the right side of the power rule gives 3log 11 a
− = .
Express 4log 7a as a product.
First rewrite 4 7 as an exponent 1
using n nx x⎛ ⎞
=⎜ ⎟⎝ ⎠
:
4l 7 o g a = .
Then use the Power Rule: 4l og 7 a = .
2 Chapter 5 Exponential Functions and Logarithmic Functions
Copyright © 2016 Pearson Education, Inc.
Express 6ln x as a product. Using the Power Rule: 6ln x = .
Example 4: Express as a difference of logarithms
8 (log By the Quotient Rule) t w=
Example 5: Express as a single logarithm
64 log 16 (By the Quotient Ru
le)
(Simplifying t he fraction)
lo
gb b− =
=
Example 6:
Express 3 5
4logax yz
in terms of sums and differences of logarithms
3 5
4
log (By the Quotient Rule)
(By the Product Ru
le)
(By th e P ower Rule )
ax yz
=
=
=
Express 2
35logaa bc
in terms of sums and differences of logarithms
13 5
4
log (Rewrite as an exponent using )
(By the Power Rule)
(By the Quotient Rule)
nnax y x xz
= =
=
=
= (By the Product Rule)
(By the Power Ru
le)
(Distri buting)
=
=
Section 5.4 Properties of Logarithmic Functions 3
Copyright © 2016 Pearson Education, Inc.
Express 5
3 4logbaym n
in terms of sums and differences of logarithms
3 5
4log (By the Quotient Rule)
(By the Pro
duct Rule)
(Distributin
g)
(By the Power R
ax yz
=
=
=
= ule)
Example 7: Express as a single logarithm
1log log (By the Power Rule)4
(By the Quo
tient Rule)
(By th
5log
e Pr oduct Rul ) e
b b bx y z− + =
=
=
Example 8: Express as a single logarithm ( ) ( )2ln 3 1 ln 3 5 2
(By the Quotient Rule) (By factoring the denominator
)
x x x+ − − −
= =
=
(By canceling 3
1) x +
These properties of logarithms can also be used to find some unknown logarithm when given some particular logarithmic values. Example 9: Given that log 2 0.301a ≈ and log 3 0.477a ≈ , find log 6a if possible.
4 Chapter 5 Exponential Functions and Logarithmic Functions
Copyright © 2016 Pearson Education, Inc.
6 (Rewriting 6 as 2 3)
(By the Product Rule)
(Substituting in the given values of log 2 and log
l
3)
(Adding)
oga
a a
= ⋅
=
≈
≈
Given that log 2 0.301a ≈ and log 3 0.477a ≈ , find 2log3a
if possible.
2 (By the Quotient Rule)3
(Substituting in the given values of log 2 and
log
lo
g 3)
(Subtracting)
a
a a
=
≈
≈
Given that log 2 0.301a ≈ and log 3 0.477a ≈ , find log 81a if possible.
4
81 (By noticing that 81 3 )
(By the Power Rule)
(Substituting in the given values of log 3)
(M
log
ultiplying )
a
a
= =
=
≈
≈
Given that log 2 0.301a ≈ and log 3 0.477a ≈ , find 1log4a
if possible.
2
1 (Using the Quotient Rule)4
(Since log 1 0, and by noticing that 4 2 )
(Using the Po
wer Rul
e)
(Substituting in the given
l
og
va
a
a
=
= = =
=
≈
lues of log 2)
(Multiplying)
a
≈
Given that log 2 0.301a ≈ and log 3 0.477a ≈ , find log 5a if possible.
Section 5.4 Properties of Logarithmic Functions 5
Copyright © 2016 Pearson Education, Inc.
5 (Writing 5 as lo 3g 2 )a = +
However, we cannot rewrite this using any of our properties.
Given that log 2 0.301a ≈ and log 3 0.477a ≈ , find log 3log 2
a
a
if possible.
log 3 (Substituting in the given values of log 2 and log 3)log 2
(Dividin
g)
aa a
a
≈
≈
Another useful properties for simplifying logarithms is given below. The Logarithm of a Base to a Power For any base a and any positive real number x
log x
a a x=
Example 10: Simplify.
8log (By th e p ropert y log ) xa aa a x== =
Simplify.
8ln (Rewriting ln as log )
(By the property l og )
e
xa
e
a x
− =
= =
Simplify.
1
3
0
log10
(Rewriting log as log )
(By the property log )
k
xa a x
=
= =
A Base to a Logarithmic Power For any base a and any positive real number x
loga xa x=
6 Chapter 5 Exponential Functions and Logarithmic Functions
Copyright © 2016 Pearson Education, Inc.
Example 11: Simplify.
4 loglog4 (By the prop erty )a xk a x= =
Simplify.
ln5
log
(Rewriting ln as log )
(By the property )a x
ee
a x
=
= =
Simplify.
10
log7
log
10
(Rewriting log as log )
(By the property ) a x
t
a x
=
= =
