Section 5.4 Properties of Logarithmic Functions
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: loga ( M ⋅ N ) = loga M + loga N
⎛ M ⎞
Quotient Rule: log a ⎜ ⎟ = log a M − log a N
⎝ N ⎠
Product Rule: log a M p = p ⋅ log a M
Example 1: Express as a sum of logarithms by using the Product Rule.
log3 ( 9 ⋅ 27 ) =
(By the Product Rule)
(By the definition of log3 )
=
Example 2: Express as a single logarithm.
log 2 p3 + log 2 q =
(By the Product Rule)
Example 3:
Express log a 11−3 as a product.
Compare this to the left side of the Power Rule: log a M p = p ⋅ log a M .
and p =
M=
Now inserting these in to the right side of the power rule gives
log a 11−3 =
.
Express log a 4 7 as a product.
First rewrite
log a 4 7 =
4
⎛
7 as an exponent ⎜ using
⎝
n
1
n
⎞
x = x ⎟ :
⎠
.
Then use the Power Rule:
log a 4 7 =
.
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Chapter 5 Exponential Functions and Logarithmic Functions
Express ln x 6 as a product.
Using the Power Rule:
ln x6 =
.
Example 4:
Express as a difference of logarithms
8
logt =
(By the Quotient Rule)
w
Example 5:
Express as a single logarithm
logb 64 − logb 16 =
=
(By the Quotient Rule)
(Simplifying the fraction)
Example 6:
Express log a
log a
x3 y 5
=
z4
(By the Quotient Rule)
=
(By the Product Rule)
=
(By the Power Rule)
Express log a
log a
x3 y 5
in terms of sums and differences of logarithms
z4
x3 y 5
=
z4
3
a 2b
in terms of sums and differences of logarithms
c5
1
(Rewrite as an exponent using
=
(By the Power Rule)
=
(By the Quotient Rule)
=
(By the Product Rule)
=
(By the Power Rule)
=
(Distributing)
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n
x = xn )
Section 5.4 Properties of Logarithmic Functions
Express log b
log a
3
ay 5
in terms of sums and differences of logarithms
m3 n 4
x3 y 5
=
z4
(By the Quotient Rule)
=
(By the Product Rule)
=
(Distributing)
=
(By the Power Rule)
Example 7:
Express as a single logarithm
1
5log b x − log b y + log b z =
4
(By the Power Rule)
=
(By the Quotient Rule)
=
(By the Product Rule)
Example 8:
Express as a single logarithm
ln ( 3 x + 1) − ln (3 x 2 − 5 x − 2 )
=
=
(By the Quotient Rule) (By factoring the denominator)
=
(By canceling 3 x + 1)
These properties of logarithms can also be used to find some unknown logarithm when given
some particular logarithmic values.
Example 9:
Given that log a 2 ≈ 0.301and log a 3 ≈ 0.477 , find log a 6 if possible.
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Chapter 5 Exponential Functions and Logarithmic Functions
log a 6 =
(Rewriting 6 as 2 ⋅ 3)
=
(By the Product Rule)
≈
(Substituting in the given values of log a 2 and log a 3)
≈
(Adding)
Given that log a 2 ≈ 0.301and log a 3 ≈ 0.477 , find log a
log a
2
=
3
2
if possible.
3
(By the Quotient Rule)
≈
(Substituting in the given values of log a 2 and log a 3)
≈
(Subtracting)
Given that log a 2 ≈ 0.301and log a 3 ≈ 0.477 , find log a 81 if possible.
log a 81 =
(By noticing that 81 = 34 )
=
(By the Power Rule)
≈
(Substituting in the given values of log a 3)
≈
(Multiplying)
Given that log a 2 ≈ 0.301and log a 3 ≈ 0.477 , find log a
log a
1
=
4
1
if possible.
4
(Using the Quotient Rule)
=
(Since log a 1 = 0, and by noticing that 4 = 2 2 )
=
(Using the Power Rule)
≈
(Substituting in the given values of log a 2)
≈
(Multiplying)
Given that log a 2 ≈ 0.301and log a 3 ≈ 0.477 , find log a 5 if possible.
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Section 5.4 Properties of Logarithmic Functions
log a 5 =
(Writing 5 as 2 + 3)
However, we cannot rewrite this using any of our properties.
Given that log a 2 ≈ 0.301and log a 3 ≈ 0.477 , find
log a 3
≈
log a 2
≈
log a 3
if possible.
log a 2
(Substituting in the given values of log a 2 and log a 3)
(Dividing)
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 a a x = x
Example 10:
Simplify.
log a a8 ==
(By the property log a a x = x)
Simplify.
ln e−8 =
=
(Rewriting ln as log e )
(By the property log a a x = x)
Simplify.
log103k
=
(Rewriting log as log10 )
=
(By the property log a a x = x)
A Base to a Logarithmic Power
For any base a and any positive real number x
a loga x = x
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Chapter 5 Exponential Functions and Logarithmic Functions
Example 11:
Simplify.
4log4 k =
(By the property a loga x = x)
Simplify.
eln 5 =
=
(Rewriting ln as log e )
(By the property a loga x = x)
Simplify.
10log 7 t
=
(Rewriting log as log10 )
=
(By the property a loga x = x)
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