Download 1. Apply the Product, Quotient, and Power

College Algebra
Chapter 4
Exponential and Logarithmic Functions
Section 4.4
Properties of
Logarithms
Concepts
1. Apply the Product, Quotient, and Power
Properties of Logarithms
2. Write a Logarithmic Expression in Expanded Form
3. Write a Logarithmic Expression as a Single
Logarithm
4. Apply the Change-of-Base Formula
Apply the Product, Quotient, and Power Properties of
Logarithms
Let b, x, and y be positive real numbers where b ≠ 1.
Product Property:
logb ( xy)  logb x  logb y
Quotient Property:
x
log b    log b x  log b y
 y
Power Property:
log b x  p log b x
p
For these exercises, assume that all variable
expressions represent positive real numbers.
Examples 1 – 3:
Use the product property of logarithms to write the
logarithm as a sum. Then simplify if possible.
1.
log  2xy 
3.
ln  3(a  b) 
2.
log 2  2xy 
Examples 4 – 6:
Use the quotient property of logarithms to write the
logarithm as a difference. Then simplify if possible.
4.
 z 
log 7  
 49 
6.
 100 
log 

 x y
5.
 a 
log 7  
 14 
Examples 7 – 9:
Apply the power property of logarithms.
7.
9.
ln  x
3
5

log x
2
8.
ln  e
5

Concepts
1. Apply the Product, Quotient, and Power Properties
of Logarithms
2. Write a Logarithmic Expression in Expanded Form
3. Write a Logarithmic Expression as a Single
Logarithm
4. Apply the Change-of-Base Formula
Example 10:
Write the expression as the sum or difference of
logarithms.
 5z 
log 3  
w
Example 11:
Write the expression as the sum or difference of
logarithms.
 ac 
log 7  
 5d 
Example 12:
Write the expression as the sum or difference of
logarithms.
 
2
ln x y
Example 13:
Write the expression as the sum or difference of
logarithms.
 3 x2 
ln 

 w z 


Example 14:
Write the expression as the sum or difference of
logarithms.
log 2
4x
yz 3
Example 15:
Write the expression as the sum or difference of
logarithms.
 64 x 2 y 
log8 
3 
 3zw 
Concepts
1. Apply the Product, Quotient, and Power Properties
of Logarithms
2. Write a Logarithmic Expression in Expanded Form
3. Write a Logarithmic Expression as a Single
Logarithm
4. Apply the Change-of-Base Formula
Example 16:
Write the logarithmic expression as a single logarithm
with a coefficient of 1, and simplify as much as possible.
log( 4 x  3)  log x
Example 17:
Write the logarithmic expression as a single logarithm
with a coefficient of 1, and simplify as much as possible.
log 2 z  4 log 2 y
Example 18:
Write the logarithmic expression as a single logarithm
with a coefficient of 1, and simplify as much as possible.
3 ln x  ln( x  2)  ln 5
Example 19:
Write the logarithmic expression as a single logarithm
with a coefficient of 1, and simplify as much as possible.
2
2


2 ln x  ln  x    ln z 9  ln z
3
Example 20:
Write the logarithmic expression as a single logarithm
with a coefficient of 1, and simplify as much as possible.
2 log 5  x  5   log5 x  log5  x 2  25 
Examples 21 – 23:
Use logb 2  0.4307, logb 3  0.6826, and logb 7  1.2091
to approximate the value of
21.
logb 21
23.
7
log b  
2
22.
logb 9
Concepts
1. Apply the Product, Quotient, and Power Properties
of Logarithms
2. Write a Logarithmic Expression in Expanded Form
3. Write a Logarithmic Expression as a Single
Logarithm
4. Apply the Change-of-Base Formula
Apply the Change-of-Base Formula
Let a and b be positive real numbers such that a ≠ 1 and
b ≠ 1. Then for any positive real number x
log b x 
log a x
log a b
In particular,
log b x 
log x
log b
or
ln x
ln b
Examples 24, 25:
Use the change-of-base formula and a calculator to
approximate the logarithm to 4 decimal places.
24.
log 7 15
25.
log5 0.3