6-2: Properties of
Logarithms
Unit 6: Exponents/Logarithms
English Casbarro
You can find these properties because of
exponential rules.
a. bmbn= bm+n
b. bm= bm-n
bn
c. (bm)n= bmn
Example 1:
a. log42 + log432
Express as a single logarithm. Simplify.
b. log5625 + log525
c. log1 27 + log1
3
3
1
9
Example 2:
a. log232 – log24
Express as a single logarithm. Simplify.
b. log749 – log77
c. log216 – log22
Example 3:
a. log3814
Express as a single logarithm. Simplify.
1 3
b. log5 ( 5 )
c. log5252
Exponential and Logarithmic are inverses, so
They “undo” each other.
For any base b, such that b > 0 and b ≠ 1
Algebra
Example
logbbx = x
log10107 = 7
blogbx= x
10log102= 2
Change of base formula
For a > 0 and a ≠ 1 and any base b, such that b > 0 and b ≠ 1
Algebra
logb x =
Note:
Example
loga x
loga b
log4 8 =
log2 8
log2 4
This is most often used to change the base to 10 or e so that
you can use your calculator.
Example 4:
a.
log8 8
d.
3 x +1
log4 8
Simplify each expression.
b. log 125
5
e.
log9 27
c.
log2 27
2
f.
log8 16
Not all logarithms involve strictly numbers; some also involve variables.
The properties work exactly the same way.
Example 5:
Write as a single logarithm.
These properties are used to evaluate expressions as well.
Example 6:
a.
c.
b.
d.
Example 7:
Solve the following problem for x using the properties.
log3 ( x 2 + 7 x - 5) = log3 (6 x +1)
Example 8:
Solve the following problem for x by using the properties.
2loga 3 + loga ( x - 4) = loga ( x + 8)