Download Logarithmic Functions

5.4 Properties of
Logarithms
3/1/2013
Properties of Logarithms
Let m and n be positive numbers and b ≠ 1,
Product Property
log b mn  log b m  log b n
Quotient Property
m
log b  log b m  log b n
n
Power Property
log b m  n log b m
n
Example 1
Expand a Logarithmic Expression
Expand the expression. Assume all variables are
positive.
a. log4
5x 2
3x
b. log7
y
SOLUTION
a. log4 5x 2 = log4 5 + log4 x 2
= log4 5 + 2 log4 x
3x
b. log7
log7 3x – log7 y
y =
= log7 3 + log7 x – log7 y
Product property
Power property
Quotient property
Product property
Checkpoint
Expand and Condense Logarithmic
Expressions
Expand the expression.
5. log2 5x
ANSWER
log2 5 + log2 x
6. log 2x 3
ANSWER
log 2 + 3 log x
5x
7. log3
7
ANSWER
log3 5 + log3 x – log3 7
4x 2
8. log6
y
ANSWER
log6 4 + 2 log6 x – log6 y
Example 2
Solve for x
Find the value of x.
a. log5 125 = x
Rewrite in Exp. Form:
5𝑥 = 125
53 = 125
x=3
(5 raised to what power equals 125?)
b. Logx 64 = 3
Rewrite in Exp. Form:
𝑥 3 = 64 (what do you raised to 3rd power to get 64?)
43 = 64
x=4
Example 2
Solve for x
Find the value of x.
c. log3 x = 4
Rewrite in Exp. Form:
34 = 𝑥
81 = 𝑥
x = 81
(3 to the 4th power equals what?)
d. ln x = 4
Rewrite in Log. Form:
log 𝑒 𝑥 = 4
Rewrite in Exp. Form:
𝑒4 = 𝑥
x = 𝑒4
e. ln x = -1
Rewrite in Log. Form:
log 𝑒 𝑥 = −1
Rewrite in Exp. Form:
𝑒 −1 = 𝑥
1
x=
𝑒
Homework:
WS 5.4 #1-7odd, 8-17 all,19-29 odd