1. Unit 11: Logarithms, day 3 3 properties to Expand and Condense Logarithmic Expressions

2. 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
log2 15 = log2 3 + log2 5 (since 3*5 = 15)

3. 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
log2 3x + log2 5y = log2 15xy (since 3x*5y = 15xy)

4. 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
log2 3/5 = log2 3 – log2 5

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
log2 3y – log2 5x = log2

6. 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 (EXPANDING)
EX:
log3 x5 = 5 log3 x
log 311 = 11 log 3

7. 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 (CONDENSING)
EX:
5 log3 x = log3 x5
w log 3 = log 3w

8. Additional Examples: Expand the logarithms (completely):
TIP: Always do PRODUCT & QUOTIENT before POWER when expanding
Expand the logarithms (completely):
log 3x2 = log 3 + log x2 (Product Property)
= log log x (Power Property)
log 4x5y7 = log 4 + log x5 + log y7 (Product)
= log log x + 7 log y (Power)
3. log = log (5y4) – log (2x3) (Quotient)
= log 5 + log y4 – log 2 – log x3 (Product)
= log log y – log 2 – 3 log x (Power)
(Why does the “3 log x” have to be subtracted?)

9. Additional Examples: Condense the logarithms (completely):
TIP: Always do POWER before PRODUCT & QUOTIENT when condensing
Condense the logarithms (completely):
log log x = log 6 + log x4 (Power Property)
= log 6x (Product Property)
log log x log y
= log 17 + log x2 + log y0.5 (Power)
= log 17x2y (Product)
3. log log w – 3 log 2 – 4 log x
= log 7 + log w2 – log 23 – log x (Power)
= log (Product & Quotient Properties)

10. PRACTICE: log4 7x log2 log x7 log5 3x2 5. log2
Write each problem on a sheet of paper and then either expand or condense…
Condense the following logarithms.
6. log 6 – log 2
7. log2 5 + log2 x
log4 x
9. log4 x + log4 y – log4 w
log3 x + 2 log3 y
Expand the following logarithms.
log4 7x
log2
log x7
log5 3x2
5. log2