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M 1310
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Laws of Logarithms
The laws:
YOU MUST LEARN THESE!
If m, n and b are positive numbers, b ≠ 1 then
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log 1 = 0
b
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logb ⎜ ⎟ = logb P − logb Q
⎝Q⎠

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Note: logb = − logb Q Why?
Q
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blogb P = P
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(Change of base formula)
logb x
loga x =
logb a
Note: this results in
ln x
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Errors to avoid:

loga (x + y) ≠ loga x + loga y
loga x
≠ loga x − loga y
loga y
(loga x)3 ≠ 3 loga x
Using the properties to simplify logarithms
Example 1: Simplify each logarithm

log2 (32)
log6 9 + log6 4
log4 ( 96 ) − log4 (6)
⎛ 1⎞
log3 ⎜ ⎟
⎝ 81 ⎠

( )

log2 46

logb

( b)

( )

⎛ 1 ⎞
loga ⎜ ⎟ − loga a2
⎝ a3 ⎠

M 1310
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log3 (x(x + 4))

Example 3: Rewrite the expression in a form with no logarithm of a
product, power, or quotient
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⎛ ab3 ⎞
ln ⎜
⎟
⎜ c2 d ⎟
⎝
⎠

M 1310
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⎛ x+5 ⎞
log3 ⎜
⎟
⎝ x2 − 4 ⎠

Example 6: Rewrite the expression in a form with no logarithm of a
product, power, or quotient
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Example 7: Rewrite the expression in a form with no logarithm of a
product, power, or quotient
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log2 35

log3 100

Combining a sum of logarithmic expressions:
Example 9: Rewrite as a single logarithm
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Example 10: Rewrite as a single logarithm
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2 ln x − 5 ln(x + 1) + 1 ln(x − 3)
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M 1310
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3 log5 (x + 2) − 2 log5 (x − 1) − 2 log5 (x − 7 )

Example 13: Rewrite as a single logarithm
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The change of base formula:
Why? The change of base formula is typically used in three situations
- When an expression or equation involves logs in two or more
different bases
- When we want to evaluate a logarithm on a calculator and need
to convert a base to base 10 or base “e” to use our calculator
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Example 14: Use the change of base formula to change log2 17 to
natural log
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