1. 5.4 Properties of Logarithms 3/1/2013

2. Properties of Logarithms
Let m and n be positive numbers and b ≠ 1,
Product Property
Quotient Property
Power Property

3. Expand the expression. Assume all variables are positive.
Example 1
Expand a Logarithmic Expression
Expand the expression. Assume all variables are positive.
log7 y 3x b. a.
log4 5x 2
SOLUTION a.
log4 5x 2 =
log log4 x 2 +
Product property =
log log4 x +
Power property
log7 y 3x b. =
log7 3x log7 y –
Quotient property
log log7 x log7 y = + –
Product property 3

4. Checkpoint Expand the expression. 5. log2 5x ANSWER log2 5 + log2 x 6.
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 7.
log3 7 5x
ANSWER
log3 5 – +
log3 x
log3 7 8.
log6 y
4x 2
ANSWER
log6 4 – +
2 log6 x
log6 y

5. 5 𝑥 =125 (5 raised to what power equals 125?) 5 3 =125 x = 3
Example 2
Solve for x
Find the value of x. a.
log5 125 = x
Rewrite in Exp. Form:
5 𝑥 =125 (5 raised to what power equals 125?)
5 3 =125
x = 3
Logx 64 = 3 b.
Rewrite in Exp. Form:
𝑥 3 =64 (what do you raised to 3rd power to get 64?)
4 3 =64
x = 4 5

6. 3 4 =𝑥 (3 to the 4th power equals what?) 81=𝑥 x = 81
Example 2
Solve for x
Find the value of x. c.
log3 x = 4
Rewrite in Exp. Form:
3 4 =𝑥 (3 to the 4th power equals what?)
81=𝑥
x = 81
ln x = 4 d.
ln x = -1 e.
Rewrite in Log. Form:
log 𝑒 𝑥 =4
Rewrite in Exp. Form:
𝑒 4 =𝑥
x = 𝑒 4
Rewrite in Log. Form:
log 𝑒 𝑥 =−1
Rewrite in Exp. Form:
𝑒 −1 =𝑥
x = 1 𝑒 6

7. Homework: WS 5.4 #1-7odd, 8-17 all,19-29 odd