Math III Unit 5: INVERSE FUNCTIONS AND
LOGARITHMS
Lauren Winstead, Heritage High School
Main topics of instruction:
1) Inverse Relations and Functions
2) Logarithmic Functions and Properties
3) The number e
4) Compounding Interest
5) Natural Logarithms
Day 1: Inverse Relations and Functions
Example 1: Find the inverse 𝑓 −1 (𝑥) of 𝑓(𝑥) = 𝑥 2 + 3.
Step 1: Graph the function.
Step 2: We find 𝑓 −1 (𝑥) by first replacing ______ with
______ and then _____________________________.
Step 3: ____________________________.
Step 4: Graph 𝑓 −1 (𝑥).
What do you notice about the graphs? They are ___________________ of each other across the
line ________________.
Is 𝑓(𝑥) a function? _________.
How do you know?
Domain
Range
____________________________.
𝑓(𝑥) =
𝑓 −1 (𝑥) =
Is 𝑓 −1 (𝑥) a function? ________.
How do you know?
____________________________.
You try!
a) Graph and find the inverse of 𝑓(𝑥) = 4𝑥 3 + 6.
Domain
𝑓(𝑥) =
𝑓 −1 (𝑥) =
Range
b) Graph and find the inverse of 𝑓(𝑥) = (𝑥 + 1)2 .
Domain
Range
𝑓(𝑥) =
𝑓 −1 (𝑥) =
Logarithmic Functions
If 𝑦 = 𝑏 𝑥 , then ______________________________.
Example 2: Convert 52 = 25 to a log function.
You try! a) Convert 43 = 64 to a log function.
4
b) Convert 1253 = 625 to a log function.
Example 3: Evaluate log 3 9 = 𝑥.
You try!
a) Evaluate log 3 81 = 𝑥.
There are two ways to do this!
Convert to exponential
Change of Base
b) Evaluate log 5 125 = 𝑥
c) Evaluate log 6 612
d) Evaluate 5log5 8
Example 4: log 3 𝑥 = 𝑦, what is the inverse of the function? Graph both.
Domain
𝑓(𝑥) =
𝑓 −1 (𝑥) =
Range
You try! If log 5 𝑥 = 𝑦, what is the inverse of the function? Graph both.
Domain
Range
𝑓(𝑥) =
𝑓 −1 (𝑥) =
Example 5: If 𝑓(𝑥) = 1.5𝑥 + 4, what is 𝑓 −1 (𝑥)?
Domain
Range
𝑓(𝑥) =
𝑓 −1 (𝑥) =
You try! If 𝑓(𝑥) = −3 + 4𝑥 , what is its inverse?
Domain
Range
𝑓(𝑥) =
𝑓 −1 (𝑥) =
Review of Transformations
Example 5: How is 𝑦 = log12 (𝑥 − 3) + 2 translated from 𝑦 = log12 𝑥?
Domain:______________ Range: __________________ Asymptote: __________________
You try! How is 𝑦 = log 5 (𝑥 + 4) − 1 translated from 𝑦 = log 5 𝑥?
Domain:______________ Range: __________________ Asymptote: __________________
Day 2: The Number e and Natural Logarithms
𝑒 ≈ _________________
The number 𝑒 is…
- ___________________________________________________________
-
___________________________________________________________
-
___________________________________________________________
Simplifying e
1) 𝑒 3 ∙ 𝑒 4 =
2)
10𝑒 3
5𝑒 2
=
3) (3𝑒 −4𝑥 )2 =
Graphing 𝑦 = 𝑒 𝑥
Graph 𝑦 = 𝑒 𝑥 .
The inverse of 𝑦 = 𝑒 𝑥 still reflects over __________,
but the function is known as a __________________
and is written as ____________________.
Domain
𝑓(𝑥) =
𝑓 −1 (𝑥) =
Range
Asymptote
Just like log functions, 𝑦 = 𝑒 𝑥 and 𝑦 = ln⁡(𝑥) can be translated up and down. Fill in the
following table.
Translations
Domain
Range
Asymptote
𝑓(𝑥) = 𝑒 −𝑥
𝑓(𝑥) = 𝑒 𝑥−2 + 1
𝑓(𝑥) = ln(𝑥 − 4) − 3
𝑓(𝑥) = ln(𝑥 + 1) + 6
Example 1: Since 𝑦 = 𝑒 𝑥 and 𝑦 = ln⁡(𝑥) are inverses, what do you think each of the
following equals?
1)
2)
3)
4)
5)
6)
ln 𝑒 4 =
ln 𝑒 28 =
ln 𝑒 =
𝑒 ln 4 =
𝑒 ln 8 =
𝑒 ln 1 =
Day 3: Properties of Logarithms
There are several properties of logarithms that allow us to simplify expressions.
Product Property
______________________________________________________________________________
Example 1: log 4 5𝑥 =
log 4 3 + log 4 𝑏 =
Quotient Property
______________________________________________________________________________
3
Example 2: ln 𝑥 =
ln 𝑎 − ln 5 =
Power Property
Example 3: log 2 7𝑥 =
3log 4 𝑥 =
Simplifying Logarithms
Example 4: Write the logarithm expression as a single logarithm.
ln 20 − ln 4
You try! Write the logarithm expressions as a single logarithm.
log 8 − 2 log 6 + log 3
Expanding Logarithms
Example 5: Expand the logarithm into multiple logarithms.
log 3 𝑥 4 𝑦 5
𝑥 2
log 5 ( )
4
You try! Expand the logarithm.
log 5 𝑎5 𝑏 7
ln
𝑥2
𝑦5
Day 4: Solving Logarithm and Exponential Equations
The key to solving any logarithmic or exponential equation is to __________________________
_____________________________________________________________________________.
Example 1: a) Solve 23𝑥 = 172.
b) log10 (3𝑥 + 1) = 5
You try! a) Solve 36𝑥 = 82
b) Solve log(7 − 2𝑥) = −1
Example 2: Solve log(7𝑥 + 1) = log(𝑥 − 2) + 1
You try! log 3 (𝑎 + 5) + log 3 (𝑎 + 2) = log 3 10
Example 3: Solve 𝑒 3𝑥 + 4 = 30
You try! Solve 6𝑒 4𝑥 − 2 = 3
Example 4: Solve 8 − ln(𝑥 − 3) = 5
You try! 12 + ln(𝑥 − 2) − ln4 = 8
Solving Systems of Equations with Logarithms
These aren’t too different from solving normal systems!
Example 1: {
You try! {
4log 𝑥 + log 𝑦 = 3
2log 𝑥 − log 𝑦 = 1
Example 2: {
You try! {
log 𝑥 + log 𝑦 = 3
log 𝑥 − log 𝑦 = 1
log 𝑥 + log 𝑦 = 2
𝑥 − 𝑦 = 21
𝑥 + 2𝑦 = 24
log 𝑥 − log 𝑦 = 1
Day 5: Finite and Continuously Compounded Interest
Often, people will invest their money at a certain percentage interest and will plan to have that
interest delivered to them in pieces throughout the year. It may be compounded once per month,
once per day, twice a year, or even constantly.
To find out how much an individual has in his or her account at any given time, we can use one
of two formulas.
Compounding a Finite Number of Times
𝑟 𝑛𝑡
𝐴 = 𝑃 (1 + )
𝑛
Example 1: I have $100 that I invest at a 5% interest rate. My interest is compounded monthly.
How much will I have after 3 years?
You try! You have $200 that you invest at a 12% interest rate compounded daily. How much
will you have after 5 years?
Compounding Continuously
𝐴 = 𝑃𝑒 𝑟𝑡
Example 2: You invest $500 to be compounded continuously throughout the year at a rate of
25%. How much money will you have after 80 years?
You try! Choose your own adventure! Decide how much you will invest, at what rate, and the
amount of time you will allow to pass. How much will you make if the interest is compounded
continuously?
Investment:
Interest Rate:
Time passed:
Try one more!
Suppose you have $4000 to deposit in a bank to earn interest for 10 years, but you have two
options. Which option would you choose and why?
Option 1: 4.5% annual interest compounded monthly
Option 2: 3.75% annual interest compounded continuously
Option 1
Option 2
Answer
Example 3: Using the same information you used above, what interest rate is required for an
investment with continuously compounded interest to triple in 5 years?
You try! What interest rate would be required for an investment with continuously compounded
interest to quadruple in 6 years?
Example 4: Suppose you deposit $3000 in an account that pays 2.5% interest compounded
quarterly. How long will it take for the account to reach $6000?
You try! You deposit $10,000 into an account that pays 3% interest compounded semiannually.
How long will it take for the account to reach $100,000?
Day 6: Advanced and Piece-wise Functions
Recall standard form for the four following types of equations:
Linear:
Quadratic:
Exponential:
Polynomial:
Which do you think moves to ∞ at the fastest rate? In other words, which is steepest?
Make up your own equations and find out! List the y values for each equation and its
corresponding x value.
x
Linear
Quadratic
0
1
2
3
4
5
6
Piece-wise Functions
Example 1: Graph the following piece-wise function:
𝑓(𝑥) = {
2𝑥 − 1⁡𝑖𝑓⁡𝑥 ≤ 1
3𝑥 + 1⁡𝑖𝑓⁡𝑥 > 1
What is 𝑓(−2)?
What is 𝑓(5)?
What is 𝑓(5) + 3𝑓(−2)?
You try! Graph the following piece-wise function:
1
3
𝑥 + ⁡𝑖𝑓⁡𝑥 < 1
𝑓(𝑥) = {2
2
−𝑥 + 3⁡𝑖𝑓⁡𝑥 ≥ 1
What is 𝑓(0)?
Exponential
Polynomial
What is 𝑓(−3)?
1
What is 4𝑓(8) − 2 𝑓(−2) + 𝑓(1)?
Example 2: Graph the following piece-wise function:
𝑓(𝑥) = {
|𝑥 + 3|⁡𝑖𝑓⁡𝑥 ≤ −2
𝑥 2 + 1⁡𝑖𝑓⁡𝑥 ≥ 0
What is the distance between the two vertices of the graph?
You try! Graph the following piece-wise function:
2𝑥 2 − 5⁡𝑖𝑓⁡𝑥 < 1
2𝑥 − 5
5
𝑓(𝑥) =
⁡𝑖𝑓⁡ ≤ 𝑥 ≤ 5
𝑥−7
2
{ log 𝑥 ⁡⁡⁡⁡𝑖𝑓⁡𝑥 ≥ 10
What is 𝑓(−2) + 3𝑓(11) − 2𝑓(0)?
Day 7: Geometric Sequences and Series
Earlier this year, you learned about arithmetic sequences, where you had to add a positive or
negative number to get to the next number in the sequence.
Geometric sequences: ___________________________________________________________
_____________________________________________________________________________.
Common ratio: ________________________________________________________________
You try! Which of the following sequences are geometric? What is the common ratio? Find
the 8th term of the geometric sequences.
1) 2, 4, 8, 16, …
2) 1, 5, 9, 13, 17,…
3) 23 , 27 , 211 , 215 , …
You probably figured that out using a recursive formula, which relies on _________________
____________________________________________________________________________.
The equation for a recursive formula looks like: _____________________________________
Example 1: Find the recursive formula for the geometric sequence 4, 12, 36, … where the a1
term is 4 and use it to find the 10th term in the sequence.
But what if I asked you to find the 50th term in a geometric sequence? That would take a lot of
work! Instead, you can use an explicit formula, which takes you right to the answer.
Explicit formula: _______________________________
Example 2: The first term of a geometric sequence is 2. The fourth term is -54. Find the
second and third terms.
You try! Given some terms in a geometric sequence, find the missing terms. Then, write the
recursive formula for the sequence.
1) 972, ___, ___, ___, 12
2) 2.5, ___, ___, ___, 202.5
3) 12.5, ___, ___, ___, 5.12
Example 3: For the geometric sequence 3, 12, 48, 192,…, find the explicit formula and the
indicated term.
1) 5th term
2) 17th term
3) 20th term
4) nth term
You try! Find the explicit formula and the 10th term of each geometric sequence:
1
1
1
1) a1 = 8, r = 2
2) a1 = − 3 , 𝑟 = ⁡ 2
Geometric series: ________________________________________________________
The sum of a finite geometric series:
Example 4: Find the sum of the finite geometric series 3 + 6 + 12 + 24 + …+ 3072.
You try! Find the sum of the finite geometric series -5 + -10 + -20 + -40 + …+ -2560
Sums of geometric series can also be expressed in __________________________________.
1
𝑛
Example 5: Find the sum of the finite geometric series ∑20
𝑛=0 4(2)
2
You try! Find the sum of the finite geometric series ∑5𝑛=1(3)𝑛−1
You can also calculate the sum of an infinite geometric series with first term a1 and common
ratio ____________. These series ________________ to a sum. (If |𝑟| > 1, there is no finite sum
for the series. These series ______________ to an infinite sum.)
1
1
Example 6: Does the series 1 + ⁡ 2 + ⁡ 4 + ⋯ converge or diverge? If it converges, what is the
sum?
2
5 𝑛
What about the series ∑∞
𝑛=0 (3) (− 4) ?
You try! Determine if the series converges or diverges, then find the sum if it converges.
a)
1
3
9
+4+8+⋯
2
2 𝑛
b) ∑∞
𝑛=1 (3)