Essential Question: What are the three properties

that simplify logarithmic expressions? Describe how to use them.

Rule: All logs must share the same base in order to be expanded/compressed

#1) Product Property◦ logb (vw) = logb v + logb w◦ Example: log7 3 + log7 11 = log7 33

#2) Quotient Property◦ logb ( ) = logb v – logb w

◦ Example: log5 28 - log5 7 = log5 4

v

w

#3) Power Property◦ log (vk) = k log v◦ Example #1: log9 45 = 5 log9 4

◦ Example #2: log8 = log8 6½ = ½ log8 6

State the property used to rewrite each expression:◦ log3 32 – log3 8 = log3 4 quotient property

◦ log6 = log6 x power property

6

n pxp

n

YOUR TURNState the property used to rewrite each expression:◦ log5 2 + log5 6 = log5 12

◦ 3 logb 4 – 3 logb 2 = logb 8Product propertyPower & Quotient property

Write each logarithmic expression as a single logarithm◦ log3 20 – log3 4

◦ 3 log2 x + log2 y

= log3 20/4 Quotient Property= log3 5 Divide

= log2 x3 + log2 y Power Property

= log2 x3y Product Property

Your Turn Write 3 log 2 + log 4 – log 16 as a single

logarithm

log 2

Logarithms can also be expanded◦ log5 x/y

◦ log 3r4

◦ log2 7b

◦ log7 a3b4

log5 x - log5 y

log 3 + 4log r

log2 7 + log2 b

3log7 a + 4log7 b

Assignment◦ Page 457◦ 1 – 29, odd problems