Properties of Logarithms
3 properties to Expand and
Condense Logarithmic
Expressions;
1 formula to Change the Base
Monday, February 8, 2016
Warm-Up
• Define the following terms: p.298 [TEXT]
– Change of Base Formula
– Product Property
– Quotient Property
– Power Property
Essential Question
• How are the properties of logarithms used to
expand, condense & evaluate logarithmic
expressions?
The Product Property
• Definition: The log of a product can be expanded
into the SUM of the logs of the factors
logb mn = logb m + logb n (EXPANDING)
EX:
log3 (7x) = log3 7 + log3 x
Note: The BASE of the
Logarithm must be and
must stay the same
when using this
property.
EX:
log2 15 = log2 3 + log2 5 (since 3*5 = 15)
The Product Property
• Definition: The SUM of logs with the same base can be
condensed into the log of the product
logb m + logb n = logb mn
(CONDENSING)
EX:
log3 7 + log3 x = log3 (7x)
EX:
log2 3x + log2 5y = log2 (15xy)… (since 3x*5y = 15xy)
The Quotient Property
• Definition: The log of a quotient can be expanded into
the DIFFERENCE of the logs of the factors
𝒎
𝐥𝐨𝐠 𝒃
𝒏
= 𝐥𝐨𝐠 𝒃 𝒎 − 𝐥𝐨𝐠 𝒃 𝒏 (EXPANDING)
EX:
log3 (7/x) = log3 7 – log3 x
EX:
log2 (3/5) = log2 3 – log2 5
A fraction can be turned into the
difference of two logs.
The Quotient Property
• Definition: The DIFFERENCE of logs with the same
base can be CONDENSED into the log of the
fraction
𝐥𝐨𝐠 𝒃 𝒎 − 𝐥𝐨𝐠 𝒃 𝒏 =
EX:
EX:
log 3 7 − log 3 2 =
𝒎
𝐥𝐨𝐠 𝒃
𝒏
(CONDENSING)
7
log 3
2
log 2 3𝑦 − log 2 5𝑥 =
3𝑦
log 2
5𝑥
The Power Property
• Definition: The log of a power expression can be
expanded into the exponent times the log of the base
𝐥𝐨𝐠 𝒃 𝒎𝒑 = 𝒑 ∙ 𝐥𝐨𝐠 𝒃 𝒎
EX:
log3 x5 = 5 log3 x
EX:
log 311 = 11 log 3
(EXPANDING)
The Power Property
• Definition: A number times the log of an expression
can be CONDENSED into the log of the expression to
the power of the number
𝒑 ∙ 𝐥𝐨𝐠 𝒃 𝒎 = 𝐥𝐨𝐠 𝒃 𝒎𝒑 (CONDENSING)
EX:
5 log3 x = log3 x5
EX:
w log 3 = log 3w
Additional Examples:
TIP: Always do
PRODUCT & QUOTIENT
before POWER when
expanding
Expand the logarithms (completely):
1. log 3x2 = log 3 + log x2 (Product Property)
= log 3 + 2 log x (Power Property)
2. log 4x5y7 = log 4 + log x5 + log y7 (Product)
= log 4 + 5 log x + 7 log y (Power)
3. log
= log (5y4) – log (2x3) (Quotient)
= log 5 + log y4 – log 2 – log x3 (Product)
= log 5 + 4 log y – log 2 – 3 log x (Power)
(Why does the “3 log x” have to be subtracted?)
Additional Examples:
TIP: Always do POWER
before PRODUCT &
QUOTIENT when
condensing
Condense the logarithms (completely):
1. log 6 + 4 log x = log 6 + log x4 (Power Property)
= log 6x4
(Product Property)
2. log 17 + 2 log x + 0.5 log y
= log 17 + log x2 + log y0.5 (Power)
= log 17x2y0.5
(Product)
3. log 7 + 2 log w – 3 log 2 – 4 log x
= log 7 + log w2 – log 23 – log x4
= log
(Power)
(Product & Quotient Properties)
The Change of Base Formula
• The Change of Base Formula can be used to
change any single logarithm into the division
of two logarithms of any desired base.
logb x = loga (x)/ loga (b)
… where “a is the desired base
Ex: log2 (7) = log (7)/log (2) … common log
or
= ln (7)/ ln(2) … natural log
or
= log5 (7)/log5 (2)… base 5
This makes evaluating logarithms of different bases easier. You can use the
LOG or LN button on your calculator… log2 (7) = log (7)/log (2) ≈ 2.8074
HOMEWORK
• Use your workbook pages 258-259 and do
problems (1 – 6)
Reflection
• What is one bit of advice you would tell
someone in another class who hasn't learned
this yet?