Unit 11: Logarithms, day 3
3 properties to Expand and
Condense 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
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
logb m/n = logb m – logb n (EXPANDING)
EX:
log3 7/x = log3 7 – log3 x
EX:
log2 3/5 = log2 3 – log2 5
The Quotient Property
• Definition: The DIFFERENCE of logs with the same base
can be CONDENSED into the log of the fraction
logb m – logb n = logb m/n (CONDENSING)
EX:
log3 7 – log3 2 = log3 7/2
EX:
log2 3y – log2 5x = log2
The Power Property
• Definition: The log of a power expression can be
expanded into the exponent times the log of the base
logb mp = p ● logb 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 ● logb m = logb mp
EX:
5 log3 x = log3 x5
EX:
w log 3 = log 3w
(CONDENSING)
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)
PRACTICE:
Write each problem on a sheet of paper and then either expand or
condense…
Expand the following
logarithms.
Condense the following
logarithms.
1.
2.
3.
4.
5.
6. log 6 – log 2
log4 7x
log2
log x7
log5 3x2
log2
7. log2 5 + log2 x
8. 6 log4 x
9. log4 x + log4 y – log4 w
10. 4 log3 x + 2 log3 y