Brief 3.1 Review
Exponential Functions
Exponential Functions
Additional Example
Use transformation of 𝑦 = 2𝑥 to graph the function.
y
𝑓 𝑥 = −2𝑥−1
5
Asymptote(s):
x
Domain:
Range:
-5
-5
5
The Natural Base 𝒆
• Irrational number
(transcendental actually . . . like 𝜋)
• 𝑒 is called the “natural base” and 𝑓 𝑥 = 𝑒 𝑥 is
called the “natural exponential function.”
• 𝑒 ≈ 2.718281827
• Formal Definition of 𝑒:
1
lim 1 +
𝑛→∞
𝑛
𝑛
Section 3.2
Logarithmic Functions
𝑥
What’s the inverse of 𝑓 𝑥 = 2 ?
y
5
x
-5
-5
5
Formal Definition
Example 1
Write each equation in its equivalent exponential
form:
a. 2 = log 5 𝑥
b. 3 = log 𝑏 64
c. log 3 7 = 𝑦
Your Turn
Write each equation in its equivalent exponential
form:
a. 3 = log 7 𝑥
b. 2 = log 𝑏 25
c. log 4 26 = 𝑦
Example 2
Write each equation in its equivalent logarithmic
form:
a. 122 = 𝑥
b. 𝑏 3 = 8
c. 𝑒 𝑦 = 9
Your Turn
Write each equation in its equivalent logarithmic
form:
a. 25 = 𝑥
b. 𝑦 9 = 216
c. 𝑒 𝑥 = 33
Example 3
Evaluate.
a. log 2 16
b.
1
log 7
49
Example 3 (cont.)
c. log 25 5
5
d. log 2 2
e.
1
log 3
3
Some Basic Properties
Example 4
Evaluate:
a. log 7 7
b. log 5 1
Remember . . .
• The logarithmic function is the inverse of the
exponential function.
f(f
1
( x))  f
1
( f ( x))  x
Example 5
Evaluate.
a. log 4 45
b. 6log6 9
Graphs of Logarithmic Functions
Asymptote(s):
Domain:
Range:
Two Cases
1. 𝑏 > 0
Two Cases (cont.)
2. 0 < 𝑏 < 1
Example 6
Use transformations to graph the function.
𝑓 𝑥 = log 2 𝑥 − 1 − 2
y
5
Asymptote(s):
Domain:
x
Range:
-5
-5
5
Example 7
Find the domain.
𝑓 𝑥 = log 4 (𝑥 + 3)
Common Logarithms
• The logarithmic function with base 10 is called
the common logarithmic function.
• Typically, log10 𝑥 is just expressed log 𝑥.
• On your calculator, hit the
button.
Example 8
Evaluate without a calculator.
a. log 1000
b. 10log 51
c. log .01
Natural Logarithms
• The logarithmic function with base 𝑒 is called
the natural logarithmic function.
• Typically, log 𝑒 𝑥 is just expressed ln 𝑥.
• On your calculator, hit the
button.
Example 9
Evaluate without a calculator.
a. ln 𝑒
b. 𝑒
𝑥+1
ln 5𝑥 2
c. log .01
Example 10
Use transformations to graph the function.
𝑓 𝑥 = ln −𝑥 − 2
y
5
Asymptote(s):
Domain:
x
Range:
-5
-5
5
Example 11
Find the domain.
𝑓 𝑥 = ln(3 − 𝑥)
Example 12
Find the domain.
𝑓 𝑥 = ln(𝑥 − 3)2
True or False?
a.
log2 8
log2 4
=
8
4
True or False
b. log −100 = −2
True or False
c. log 𝑏 𝑥 is the exponent to which
𝑏 must be raised to get 𝑥.
True or False
Questions???
Make sure to be working in MyMathLab!!!