Laws of Logarithms
MCR3U9
Important properties of logarithms:
log 𝑏 1 =
log 𝑏 𝑏 𝑥 =
log 𝑏 𝑏 =
𝑏 log𝑏 𝑥 =
Since logarithms are exponents, the laws of exponents give corresponding laws of logarithms.
Law of Logarithms
Equivalent Exponent Rule
Product Law of Logarithms
log 𝑏 (𝑀𝑁) = log 𝑏 𝑀 + log 𝑏 𝑁
Quotient Law of Logarithms
𝑀
log 𝑏 ( ) = log 𝑏 𝑀 − log 𝑏 𝑁
𝑁
Power Law of Logarithms
log 𝑏 𝑥 𝑛 = 𝑛 log 𝑏 𝑥
Examples of how to use each law of logarithm:
Example 1
Evaluate.
a) log 4 2 + log 4 32
b) log 2 144 − log 2 9
c) log 0.0012
The power law is also used to evaluate logs with bases other than 10.
Example 2
Evaluate 2 = 1.05𝑥
Change of Base Formula
Many calculators cannot calculate logs of bases other than 10 or 𝑒. We can use the change of
base formula to calculate logarithms of different bases.
log 𝑛 𝑚 =
logb 𝑚
logb 𝑛
𝑏 > 0,
𝑏 ≠ 1,
𝑚 > 0,
𝑛>0
Example 3
Evaluate log 2 41 correctly to 3 decimal places.
Example 4
Evaluate each and state the rule(s) used.
4
a) log 3 9
b) log 2 85
c) log 5 125
e)
log 8 4  log 8 16
h) log 7
7
 log 7 56
8
d) log 4 10
f) log 3 405  log 3 5
1
g) 2 log 5  log 16
2
i) log 2 144  log 2 9
j) log 3 27 3
Example 5 Write as a single log
a) log 5 6  log 5 8  log 5 16
b) 3 log 3 2  log 3 4
c) 2 log 2 32  log 2 16  3 log 2 3
d) log 6 3 + 2 log 6 5 − log 6 2
e) log 𝑎 𝑏 + log 𝑎 (7𝑐) + log 𝑎 (4𝑏) − log 𝑎 𝑐
1