1. SOLVING (expand and condense)
3.3: Properties of Logs
SOLVING (expand and condense)

2. 3.3 Properties of Logarithms
Essential Questions:
How can I use the properties of logarithms to expand, condense, and solve logarithmic equations?

3. 1st Type of Logarithm: Common Log  𝐥𝐨𝐠 𝒙 =𝒚 Examples:
If there is no “b” term, the log has base 10.
Examples:
log 45 = ____ log = ____ log ¾ = _____
1.65
.507
-.125

4. 2nd type of Logarithm: Natural Log  𝐥𝐧 𝒙 =𝒚 ln 4 loge4 = ___________
ln implies a base of “e”
ln 4
loge4 = ___________
Examples:
Ln 5 = _____ Ln 3.144= _____ Ln 5/8 = _____
1.61
1.45
-.47

5. Laws of Natural Logarithms
ln XY =
ln X + ln Y
Product: Quotient: Power: One-to-One: Identity:
ln 𝑋 𝑌 =
ln X – ln Y
ln XY =
Y ln X
If ln X= ln Y,
then X = Y
If ln ex
= x

6. Expand the logarithmic expression:
a) ln 7 𝑦 3 4 𝑥 3
ln ln y – ln 4 – 3 ln x OR
ln ln y – (ln ln x)
b) ln 6 𝑥 2 𝑦 4
ln ln x – 4 ln y

7. c) ln 6 𝑏
(1/2)ln6 – (1/2)lnb
d) ln 4w 3 𝑥 2
ln4 + lnw + (2/3)lnx

8. Condense to a single logarithm
a) 5 ln(x + 1) + ln x b) 6 ln(x – 4) + 3 ln x c) ln (3x + 5) – 4ln x – 6ln(x – 1)
ln x(x + 1)5
ln x3(x – 4)6
ln 3𝑥+5 𝑥 4 (𝑥−1 ) 6

9. One-to-One and Identity equations
Solve for x.
a) ln 3 𝑥 2 = ln 4 b) ln ex+1 = 4
3 𝑥 2 =4
x + 1 = 4
x = 3
x2 = 64
x = ± 8

10. Homework 3.3 Properties of Logs Worksheet
“No, not my dog. I do my homework on my computer…and the cat ate the mouse.