This document discusses the four basic properties of logarithms: the product rule, quotient rule, power rule, and examples of applying each rule to write logarithmic expressions as single logarithms or expand logarithmic expressions. More complex examples show condensing multiple logarithms using properties and expanding a logarithm using the rules.

2. LOGARITHMIC
PROPERTY DAY!!!!
8.4 – Properties of Logarithms

3. Properties of Logarithms

There are four basic properties of
logarithms that we will be working with.
For every case, the base of the logarithm
can not be equal to 1 and the values must
all be positive (no negatives in logs)

4. Product Rule
logbMN = LogbM + logbN

Ex: logbxy = logbx + logby

Ex: log6 = log 2 + log 3

Ex: log39b = log39 + log3b

5. Quotient Rule
log



Ex: log
Ex: log
Ex: log
M
b
5
MN
2
P
M
log
5
x
log
log
2
a
log
y
a
2
b
N
x
5
log
log
2
M
log
b
N
y
5
2
5
log
2
N
log
2
P

6. Power Rule
log

Ex: log

Ex:

Ex: log
5
log
2
B
2
5
x
3
7
M
b
a b
4
x
x log
2 log
x log
3 log
5
b
M
B
2
5
7
a
4 log
7
b

7. Let’s try some

Working backwards now: write the following as a single
logarithm.
log 4 4
log 4 16
log 5
log 2
2 log 2 m
4 log 2 n

8. Let’s try some

Write the following as a single logarithm.
log 4 4
log 4 16
log 5
log 2
2 log 2 m
4 log 2 n

9. Let’s try something more
complicated . . .
Condense the logs
log 5 + log x – log 3 + 4log 5
log 4 5
2 log 4 x
5 (log 4 3 x
log 4 5 x )

10. Let’s try something more
complicated . . .
Condense the logs
log 5 + log x – log 3 + 4log 5
log 4 5
2 log 4 x
5 (log 4 3 x
log 4 5 x )

11. Let’s try something more
complicated . . .

Expand
log
10 x
3y
3
4
2
log
2 x
8
5

12. Let’s try something more
complicated . . .

Expand
log
10 x
3y
3
4
2
log
2 x
8
5