M 1310
5.4
1
You will sometimes be asked to rewrite logarithmic expressions
in either an expanded or contracted form. To do this, you will use
the Laws of Logarithms. You will need to know all 8 of these
laws.
Laws (properties, rules) of logarithms:
If m, n and b are positive numbers, b ≠ 1 then
1. log b = 1
b
2. log 1 = 0
b
3. log (PQ ) = log P + log Q
b
b
b
P
4. logb   = logb P − logb Q
Q
Note: logb
1
= − logb Q Why?
Q
5. log P n = n log P
b
b
6. b logb P = P
7. log b P = P
b
(Change of base formula)
logb x
loga x =
logb a
Note: this results in
ln x .
loga x =
ln a
Examples using these properties:
8.
M 1310
5.4



Errors to avoid:
loga (x + y ) ≠ loga x + loga y
loga x
≠ loga x − loga y
loga y
 (loga x )3 ≠ 3 loga x
Putting the properties together:
Example 1:
Rewrite the expression in a form with no
logarithm of a product, power, or quotient.
log3 (x (x + 4 ))
Example 2:
Rewrite the expression in a form with no
logarithm of a product, power, or quotient.
log
3
x
Example 3:
Rewrite the expression in a form with no
logarithm of a product, power, or quotient.
 ab 3 

ln
 c2 d 


2
M 1310
5.4
Rewrite the expression in a form with no
Example 4:
logarithm of a product, power, or quotient.
 x +5 
log3 

 x2 − 4 
Example 5:
Rewrite the expression in a form with no
logarithm of a product, power, or quotient.
 x 2 (x + 1) 
log10 

4
 (x − 3 )(x + 7) 
Example 6:
Rewrite the expression in a form with no
logarithm of a product, power, or quotient.
ln
(x − 3)(x + 9)3
x 5 (x − 10 )
3
M 1310
Example 7:
5.4
Rewrite as a single logarithm.
log3 x + log3 2
Example 8:
Rewrite as a single logarithm.
loga b − c loga d + r loga s
Example 9:
Rewrite as a single logarithm.
2 ln x − 5 ln(x + 1) +
1
ln(x − 3)
2
4
M 1310
Example 10:
5.4
5
Rewrite as a single logarithm.
3 log5 (x + 2 ) − 2 log5 (x − 1) − 2 log5 (x − 7)
Example 11: Use the change of base formula to change log2 17 to
natural log.
Example 12: Use the change of base formula to change log7 12 to
base 10.