12.3A Properties of Logarithms
Since logarithms are exponents, we should expect logarithms to
have properties that correspond to the properties of exponents.
𝒃𝒎 ∙ 𝒃𝒏 = 𝒃𝒎+𝒏
𝒃𝒎
𝒃𝒏
Product Rule
= 𝒃𝒎−𝒏
Quotient Rule
(𝒃𝒎 )𝒏 = 𝒃𝒎𝒏
Power-of-a-Power Rule
Expanding Logarithms
The following three rules are used to expand logarithms.
m and n are any real number, variable or expression >0.
b>0 and b≠1. c is any real number.
Product Rule
𝒍𝒐𝒈𝒃 𝒎𝒏 = 𝒍𝒐𝒈𝒃 𝒎 + 𝒍𝒐𝒈𝒃 𝒏
Quotient Rule 𝒍𝒐𝒈𝒃
Power Rule
𝒎
𝒏
= 𝒍𝒐𝒈𝒃 𝒎 − 𝒍𝒐𝒈𝒃 𝒏
𝒍𝒐𝒈𝒃 𝒎𝒄 = 𝒄 ∙ 𝒍𝒐𝒈𝒃 𝒎
NOTE: There is no property of logarithms that can be used to
simplify the log of a sum, 𝒍𝒐𝒈𝒂 (𝒎 + 𝒏), or the log of a
difference, 𝒍𝒐𝒈𝒂 (𝒎 − 𝒏).
To expand a logarithm, use the rules of logarithms to rewrite the
logarithmic expression until all arguments are prime or
simplified to a rational value.
Expand 𝒍𝒐𝒈 𝟑𝒙
The argument, 3x, is a product. Use the Product Rule.
log 3x = log 3 + log x
all arguments are prime
Expand 𝒍𝒐𝒈𝟓 𝟔
The argument, 6, is not prime. Write it as a product of primes,
𝟔 = 𝟐 ∙ 𝟑. Use the Product Rule.
𝒍𝒐𝒈𝟓 𝟔 = 𝒍𝒐𝒈𝟓 (𝟐 ∙ 𝟑) = 𝒍𝒐𝒈𝟓 𝟐 + 𝒍𝒐𝒈𝟓 𝟑
-------------------------------------------------------------------------------Expand 𝒍𝒐𝒈 𝟏𝟎𝒙
The argument, 𝟏𝟎𝒙, is a product. Although 10 is not prime, we do
not factor it because the log is base–10. Use the Product Rule.
𝒍𝒐𝒈 𝟏𝟎𝒙 = 𝒍𝒐𝒈 𝟏𝟎 + 𝒍𝒐𝒈 𝒙 = 𝟏 + 𝒍𝒐𝒈 𝒙
-------------------------------------------------------------------------------Expand 𝒍𝒐𝒈𝟓 𝟒𝟗
The argument, 49, is not prime. Write it as a power, 𝟒𝟗 = 𝟕𝟐 .
Use the Power Rule.
𝒍𝒐𝒈𝟓 𝟒𝟗 = 𝒍𝒐𝒈𝟓 𝟕𝟐 = 𝟐 ∙ 𝒍𝒐𝒈𝟓 𝟕
the argument, 7, is prime
-------------------------------------------------------------------------------𝒙
Expand 𝒍𝒐𝒈𝟐
𝟑
𝒙
The argument, , is a quotient. Use the Quotient Rule.
𝟑
𝒙
𝒍𝒐𝒈𝟐 = 𝒍𝒐𝒈𝟐 𝒙 − 𝒍𝒐𝒈𝟐 𝟑 all arguments are prime
𝟑
-------------------------------------------------------------------------------Expand 𝒍𝒐𝒈𝟕 𝒙𝟑
The argument, 𝒙𝟑 , is a power. Use the Power Rule.
𝒍𝒐𝒈𝟕 𝒙𝟑 = 𝟑 ∙ 𝒍𝒐𝒈𝟕 𝒙
the argument, x, is prime
If more than one rule must be applied, use the reverse order of
operations to expand the logarithm.
Expand 𝒍𝒐𝒈𝟑 𝟓𝒂𝟒
The argument, 𝟓𝒂𝟒 , contains a product and a power.
To simplify 𝟓𝒂𝟒 we would (1) do the power
(2) do the product
To expand, use the Product Rule then the Power Rule.
𝒍𝒐𝒈𝟑 𝟓𝒂𝟒 = 𝒍𝒐𝒈𝟑 𝟓 + 𝒍𝒐𝒈𝟑 𝒂𝟒 = 𝒍𝒐𝒈𝟑 𝟓 + 𝟒 ∙ 𝒍𝒐𝒈𝟑 𝒂
all arguments are prime
-------------------------------------------------------------------------------Expand 𝒍𝒐𝒈𝟑 (𝟓𝒂)𝟒
The argument, (𝟓𝒂)𝟒 , contains a product and a power.
To simplify (𝟓𝒂)𝟒 we would (1) do the product
(2) do the power
To expand, use the Power Rule then the Product Rule.
𝒍𝒐𝒈𝟑 (𝟓𝒂)𝟒 = 𝟒 ∙ 𝒍𝒐𝒈𝟑 𝟓𝒂 = 𝟒(𝒍𝒐𝒈𝟑 𝟓 + 𝒍𝒐𝒈𝟑 𝒂) =
𝟒 ∙ 𝒍𝒐𝒈𝟑 𝟓 + 𝟒 ∙ 𝒍𝒐𝒈𝟑 𝒂
all arguments are prime
-------------------------------------------------------------------------------Expand 𝒍𝒐𝒈𝟑 √𝟑𝒙
𝟏⁄
𝟐,
The argument, √𝟑𝒙 = (𝟑𝒙)
contains a product and a power.
𝟏⁄
𝟐
To simplify (𝟑𝒙)
we would (1) do the product
(2) do the power
To expand, use the Power Rule then the Product Rule.
𝟏⁄
𝟐
𝒍𝒐𝒈𝟑 √𝟑𝒙 = 𝒍𝒐𝒈𝟑 (𝟑𝒙)
𝟏
𝟏
𝟐
𝟐
= ∙ 𝒍𝒐𝒈𝟑 𝟑𝒙 = (𝒍𝒐𝒈𝟑 𝟑 + 𝒍𝒐𝒈𝟑 𝒙) =
𝟏
𝟏
𝟏
𝟏
∙ 𝒍𝒐𝒈𝟑 𝟑 + ∙ 𝒍𝒐𝒈𝟑 𝒙 = + ∙ 𝒍𝒐𝒈𝟑 𝒙
the argument, x, is
𝟐
𝟐
𝟐
𝟐
prime
See Examples 1 – 4 on pages 845 – 848.
-------------------------------------------------------------------------------Proofs of the Logarithm Properties
Product Rule
𝒍𝒐𝒈𝒃 𝒎𝒏 = 𝒍𝒐𝒈𝒃 𝒎 + 𝒍𝒐𝒈𝒃 𝒏
m is a power of a, say 𝒎 = 𝒃𝒙 → 𝒙 = 𝒍𝒐𝒈𝒃 𝒎
n is a power of a, say 𝒏 = 𝒃𝒚 → 𝒚 = 𝒍𝒐𝒈𝒃 𝒏
𝒍𝒐𝒈𝒃 𝒎𝒏 = 𝒍𝒐𝒈𝒃 𝒃𝒙 𝒃𝒚 = 𝒍𝒐𝒈𝒃 𝒃𝒙+𝒚 = 𝒙 + 𝒚 = 𝒍𝒐𝒈𝒃 𝒎 + 𝒍𝒐𝒈𝒃 𝒏
-------------------------------------------------------------------------------Quotient Rule
𝒍𝒐𝒈𝒃
𝒎
𝒏
= 𝒍𝒐𝒈𝒃 𝒎 − 𝒍𝒐𝒈𝒃 𝒏
m is a power of a, say 𝒎 = 𝒃𝒙 → 𝒙 = 𝒍𝒐𝒈𝒃 𝒎
n is a power of a, say 𝒏 = 𝒃𝒚 → 𝒚 = 𝒍𝒐𝒈𝒃 𝒏
𝒍𝒐𝒈𝒃
𝒎
𝒏
= 𝒍𝒐𝒈𝒃
𝒃𝒙
𝒃𝒚
= 𝒍𝒐𝒈𝒃 𝒃𝒙−𝒚 = 𝒙 − 𝒚 = 𝒍𝒐𝒈𝒃 𝒎 − 𝒍𝒐𝒈𝒃 𝒏
-------------------------------------------------------------------------------Power Rule
𝒍𝒐𝒈𝒃 𝒎𝒄 = 𝒄 ∙ 𝒍𝒐𝒈𝒃 𝒎
m is a power of a, say 𝒎 = 𝒃𝒙 → 𝒙 = 𝒍𝒐𝒈𝒃 𝒎
𝒍𝒐𝒈𝒃 𝒎𝒄 = 𝒍𝒐𝒈𝒃 (𝒃𝒙 )𝒄 = 𝒍𝒐𝒈𝒃 𝒃𝒄𝒙 = 𝒄𝒙 = 𝒄 ∙ 𝒍𝒐𝒈𝒃 𝒎