3.1 & 3.2 Overview
Exponential &
Logarithmic Functions
Exponential Functions
Formal Definition
Two Cases
1. 𝑏 > 0
Two Cases (cont.)
2. 0 < 𝑏 < 1
Section 3.3
Properties of Logarithms
Properties of Logarithms
For 𝑀 > 0 and 𝑁 > 0:
1. log 𝑏 𝑀𝑁 = log 𝑏 𝑀 + log 𝑏 𝑁
2.
𝑀
log 𝑏
𝑁
= log 𝑏 𝑀 − log 𝑏 𝑁
3. log 𝑏 𝑀𝑝 = 𝑝 ∙ log 𝑀
Product Rule
Quotient Rule
Power Rule
Example 1
Use properties of logarithms to expand the
logarithmic expression. Where possible, evaluate
without a calculator.
a. log 3 (13 ∙ 2)
b. log 3 (9𝑥)
c. log(1000𝑥)
True of False? (always)
log 𝑥 + 𝑦 = log 𝑥 + log 𝑦
True or False
log 𝑥 ∙ log 𝑦 = log(𝑥𝑦)
True or False
log(𝑥𝑦) = log 𝑥 + log 𝑦
True or False
Example 2
Use properties of logarithms to expand the
logarithmic expression. Where possible, evaluate
without a calculator.
a.
81
log 9
𝑦
b.
𝑒9
7
ln
c. log
𝑥
1000
True of False? (always)
log 𝑥 − 𝑦 = log 𝑥 − log 𝑦
True or False
𝑥
log
= log 𝑥 − log 𝑦
𝑦
True or False
log 𝑥
𝑥
= log
log 𝑦
𝑦
True or False
Example 3
Use properties of logarithms to expand the
logarithmic expression. Where possible, evaluate
without a calculator.
a. log 𝑏 𝑥 3
b.
log 𝑁
5
−6
c. ln 𝑥
Example 4
(same instructions as previous examples)
2
a. log 𝑏 (𝑥 𝑦)
b.
log
𝑥
100
Example 5
(same instructions as previous examples)
a. log 8
b.
log
64
𝑥+1
𝑥3𝑦
𝑧2
Example 6
(same instructions as previous examples)
log 5
3
𝑥 2𝑦
25
Example 7
Use properties of logarithms to condense each
logarithmic expression. Where possible, evaluate
without a calculator.
a. log 2 12 − log 2 6
b. log 250 + log 4
Example 8
(same instructions as Example 7)
7 log 𝑏 𝑥 + 9 log 𝑏 𝑧
Example 9
(same instructions as Example 7)
1
(log 8 𝑥 + log 8 𝑦) − 2 log 8 (𝑥 + 8)
2
Example 10
(same instructions as Example 7)
log 𝑥 + log 𝑥 2 − 49 − log 6 − log(𝑥 + 7)
Change of Base Formula
• Can be used to compute logarithms that aren’t
base 10 or base 𝑒.
So for calculator purposes . . .
• We need to apply the change of base properties
to the bases that our calculator uses.
Example 11
Use common or natural logs and a calculator
to evaluate the expression.
a. log14 16
b. log 𝜋 78
True or False?
a. ln 8𝑥 3 = 3 ln(8𝑥)
True or False
b. 𝑥 log 10𝑥 = 𝑥 2
True or False
c. ln 𝑥 + ln 2𝑥 = ln(3𝑥)
True or False
Example 12
Expand.
log
3
100𝑥 3 5−𝑥
3(𝑥+7)2
Questions???
Make sure to be working in MyMathLab!!!