* Use the product rule for logarithms. * Use the quotient rule for logarithms. * Use the power rule for logarithms. * Expand logarithmic expressions. * Condense logarithmic expressions. * Use the change-of-base formula for logarithms.

1. 6.5 Logarithmic Properties
Chapter 6 Exponential and Logarithmic Functions

2. Concepts & Objectives
⚫ Objectives for this section are
⚫ Use the product rule for logarithms.
⚫ Use the quotient rule for logarithms.
⚫ Use the power rule for logarithms.
⚫ Expand logarithmic expressions.
⚫ Condense logarithmic expressions.
⚫ Use the change-of-base formula for logarithms.

3. Properties of Logarithms
⚫ Because logarithms are exponents, they have three
properties that come directly from the corresponding
properties of exponentiation:
Exponents Logarithms
a b a b
x x x +
=
a
a b
b
x
x
x
−
=
( )
b
a ab
x x
=
( )
log log log
a b a b
= +
log log log
a
a b
b
 
= −
 
 
log log
b
a b a
=

4. Examples
1. Write log224 – log28 as a single logarithm of a single
argument.
2. Use the Log of a Power Property to solve 0.82x = 0.007.

5. Examples
1. Write log224 – log28 as a single logarithm of a single
argument.
2. Use the Log of a Power Property to solve 0.82x = 0.007.
2 2 2
24
log 24 log 8 log
8
 
− =  
 
2
log 3
=
2
log0.8 log0.007
x
=
2 log0.8 log0.007
x =
log0.007
11.12
2log0.8
x = 
=
2
0.8 0.007
x

6. Change-of-Base Formula
⚫ Although the calculators we use in class, as well as
Desmos, will calculate a logarithm using any base, it can
sometimes be useful to change a logarithm from one
base to another.
For any positive real numbers M, b, and n,
where n  1 and b  1,
=
log
log
log
n
b
n
M
b

7. Examples
1. Change log53 from base 5 to base 10.
2. Change log0.58 to a quotient of natural logarithms.

8. Examples
1. Change log53 from base 5 to base 10.
Applying the formula: M = 3, b = 5, and n = 10:
2. Change log0.58 to a quotient of natural logarithms.
Now M = 8, b = 0.5, and n = e:
10
5
10
log 3 log3
log 3 or
log 5 log5
=
0.5
ln8
log 8
ln0.5
=

9. Classwork
⚫ College Algebra 2e
⚫ 6.5: 4-14 (even); 6.3: 26-40 (even); 6.1: 62-68 (even)
⚫ 6.5 Classwork Check
⚫ Quiz 6.3