Algebra III Academic
Unit V: Exponential Functions and Logarithmic Functions
Goal




Recognize and evaluate exponential functions with base a.
Graph exponential functions and use the one-to-one property.
Recognize, evaluate, and graph exponential functions with base e.
Use exponential functions to model and solve real-life problems.
Key Vocabulary
 Base
 Common Logarithms
 Compounding
 Compounding Continuously
 Compounding Interest
 Transformation




Decay
Exponential Function
Logarithmic Function
Natural Base, e



Natural Logarithmic
Function
One-to-One Property
Properties of Logarithms
Lesson 2: Logarithmic functions and Their Graphs
Determine if the following functions are the same
1) f(x) = 3x – 2
g(x) = 3x – 9
1
h(x) = 9(3x)
2)
f(x) = 4x + 12
g(x) = 22x+6
h(x) = 64(4x)
3. Which functions are exponential?
a)
y = 3x
b) y = 3x2
c) y = 3x
D) y = 2-x
4) A philanthropist (let’s call her “M”) deposits $5000 in a trust fund that pays 7.5% interest compounded
continuously. The balance will be given to the college from which she graduated after the money has earned
interest for 50 years. How much will the college receive?
What is a logarithm?
 A logarithm is an exponent. For this reason, exponential expressions can be helpful in
understanding logarithmic ones.
o Consider the question of a square root. It helped us get comfortable with the fact that √25 is
5 to think about the fact that 52 = 25. So, it is not unprecedented to use an inverse function
to understand a function.

In general, 𝑏 𝑥 = 𝑝 can be written as 𝑙𝑜𝑔𝑏 𝑝 = 𝑥.

𝑙𝑜𝑔5 125. is read “__________________________________________”.
It asks “____________________________________________________________”
Example 1: Evaluate each logarithm for the given value of x
a) f(x) = log2x , x = 32
b)
f(x) = log3x, x = 1
c) f(x) = log4x. x = 2
d)
f(x) = log10x, x = 1/100
The Common log is ___________________________________
Example 2: Use a calculator to evaluate the function f(x) = logx at each value of x
a) x = 10
b) x = 0
c) x = -1
d) x = 2.5
e) x = 1
Properties of Logarithms
1.
2.
3.
4.
5.
loga1 = 0 because a0 = 1
logaa = 1 because a1 = a
logaax = x because ax = ax
alogax = x
If logax = logay, then x = y
Example 3: Use the properties of logs to simplify
a) log41
b)
log77
c)
6log620
Example 4: Solve for x
a) log3x = log312
b) log(2x + 1) = log 3x
c) log4(x2 – 6) = log410
When graphing exponents,
we kept track of:
1.
When graphing logs,
we will keep track of:
1.
2.
2.
3.
3.
Example 5: Graph the following. Give the equation of the vertical asymptote.
a) f(x) = 2x and g(x) = log2x
b) f(x) = logx
Example 6: Transformations of the Logarithmic Function
Describe the graph as a transformation of f(x) = log x and sketch the graph. Give the equation of the
vertical asymptote and the domain.
a) g(x) = -1 + logx
b) h(x) = log(x + 3)
HW: p. 380 1 – 6, 7-35 odd
Algebra III Academic
Unit V: Exponential Functions and Logarithmic Functions
Goal
 Use the method of substitution to solve systems of linear equations in two variables
 Use the method of substitution to solve systems of nonlinear equations in two variables
 Use a graphical approach to solve systems of equations in two variables
 Uses systems of equations to model and solve real-life problems
Key Vocabulary
 Base
 Common Logarithms
 Compounding
 Compounding Continuously
 Compounding Interest
 Transformation




Decay
Exponential Function
Logarithmic Function
Natural Base, e



Natural Logarithmic
Function
One-to-One Property
Properties of Logarithms
Lesson 3: Logarithmic functions and Their Graphs
Recall:
 Logarithmic and exponential equations are related:

The three main traits of a log. graph are:
1.
2.
3.

The Common log is _________________________________________
New Idea:
 The Natural log is ___________________________________________

The graph of the natural log will have the following traits:
1.
2.
3.
Example 7: Evaluate f(x) = ln x at each value of x
a) x = 2
b) x = 0.3
c) x = -1
d) x = 1 + √2
Properties of Natural Logarithms
1.
2.
3.
4.
5.
Ln 1 = 0 because e0 = 1
Ln e = 1 because e1 = e
Ln ex = x because ex = ex
elnx = x
If ln x = ln y then x = y
Example 8: Use the properties of the natural log to simplify
1
a) ln( )
𝑒
b) eln5
c)
𝑙𝑛1
3
d) 2lne
Example 9: Find the domain of the logarithmic functions below
a) f(x) = ln(x – 2)
b) g(x) = ln(2 – x)
c) h(x) = ln x2
d) k(x) = log x
Example 10: Application
Students participating in a psychology experiment attended several lectures on a subject and took an exam.
Every month for a year after the exam, the students took a retest to see how much of the material they
remembered. The average scores for the group are given by the human memory model f(t) = 75 – 6ln(t + 1),
where 0 ≤ t ≤ 12 where t is the time in months.
a) What was the average score on the original exam?
b) What was the average score at the end of 2 months?
c) What was the average score at the end of 6 months?
HW: p. 380 37 – 40, 41 – 75 odd, 77, 79