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