1. 8.4 – Properties of Logarithms

2. 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)

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

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

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

6. 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

7. 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

8. Let’s try something more
complicated . . .
Condense the logs
log 5 + log x – log 3 + 4log 5
log4 5 − 2 log4 x + 5(log4 3x − log4 5x)

9. Let’s try something more
complicated . . .
Condense the logs
log 5 + log x – log 3 + 4log 5
log4 5 − 2 log4 x + 5(log4 3x − log4 5x)

10. Let’s try something more
complicated . . .

Expand
4
10 x
log
3y2
2 x 

log8 
 5 


3

11. Let’s try something more
complicated . . .

Expand
4
10 x
log
3y2
2 x 

log8 
 5 


3