UNIT 5
Exponential and Logarithmic Functions!
OH YEAH!
Unit Essential Question
• How are exponential and logarithmic functions related,
and how can they be represented graphically?
LESSON 5.1
Exponential Functions
Lesson Essential Question (LEQ)
• What is an exponential function and how can they be
represented graphically?
Solving Exponential Equations
• Examples:
• 1) 43𝑥 = 46
• 2) 53𝑥 = 57𝑥−8
• 3) 32𝑥−4 = 92𝑥+6
• 4)
2 −9
𝑥
2
=4
3
2𝑥+2
Sketching Graphs
• Let’s sketch the graph of the following functions:
• Ex: 𝑓 𝑥 = 3 𝑥
• Ex: 𝑓 𝑥 = 3 𝑥−2
• Ex: 𝑓 𝑥 = 3 𝑥 + 1
• Ex: 𝑓 𝑥 = 2𝑥
1
2
• Ex: 𝑓 𝑥 = ( ) 𝑥
Even More Graphs!
• Let’s sketch the graph of:
• Ex: 𝑓 𝑥 =
• Ex: 𝑓 𝑥 =
2
𝑥
2
2
−𝑥
2
Real World Applications:
• Compound Interest
• Decay/Growth
• Half-Life
• Bloodstream
• Appreciation/Depreciation
• Inflation
• Epidemics
• Many more…
Homework:
• Page 334-335
• #’s 1, 3, 5, 7, 9, 13, 15, 17, 21, 30, 31a, 33a, 35
Bell Work:
• 1) Solve for x.
82𝑥−5
=
1 𝑥 2 −1
( )
2
Compound Interest:
• Blue Table on Page 332
𝑟
𝑛
• 𝐴 = 𝑃(1 + )𝑛𝑡
• A = Future Value
• P = Principal
• r = interest rate as a decimal
• n = number of interest periods per year
• t = number of years Principal is invested
Examples:
• Ex: If you invested $2,000 dollars 10 years ago at 4.5%
that was compounded quarterly, what would be the value
of that investment today?
• Ex: Mr. Kelsey needs to have $2,000,000 by the time he
retires in 33 years. He plans to invest in a money market
account that will return 5.25% per year. How much
money will he need to invest right now to reach his goal?
Classwork/Homework:
• Pages 335-336
• #’s 37 – 42, 45-48
Bell Work:
• The current average cost of gasoline per gallon in PA is
$2.79 and has been increasing at an average inflation
rate of 3.75% per year. If this pattern holds true, what will
be the cost of gas in 30 years?
LESSON 5.2
The Natural Exponential Function
LESSON ESSENTIAL QUESTION
• What is the natural exponential function and how can it be
used?
Important:
1
𝑛
• (1 + )𝑛 = 𝑒 = 2.71828 … 𝑎𝑠 𝑛 𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ𝑒𝑠 ∞
• How do we get this????
• The NATURAL EXPONENTIAL FUNCTION: 𝑓 𝑥 = 𝑒 𝑥
Continuously Compounded Interest
• 𝐴 = 𝑃𝑒 𝑟𝑡
• A = Future Value
• P = Principal
• r = interest rate as a decimal
• t = number of years Principal is invested
Law of Growth/Decay Formula
• 𝑞 𝑡 = 𝑞0 𝑒 𝑟𝑡
• 𝑞 𝑡 = 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦
• 𝑞0 = 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝐴𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑞
• 𝑟 = 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑔𝑟𝑜𝑤𝑡ℎ 𝑜𝑟 𝑑𝑒𝑐𝑎𝑦 𝑎𝑠 𝑎 𝑑𝑒𝑐𝑖𝑚𝑎𝑙
• 𝑡 = 𝑡𝑖𝑚𝑒 𝑖𝑛 𝑦𝑒𝑎𝑟𝑠
• If r > 0, then the quantity is growing.
• If r < 0, then the quantity is decaying.
Homework:
• Pages 345-346
• #’s 5 – 15 odds, 19-31 odds
Bell Work:
• 1) If $12,000 is continuously compounded at 2.55% over
5 years and 3 months, what would be the amount of
interest earned?
• 2) The population of PA in 1990 was 12 million. If the
population is continuously growing at a rate of 1.05% per
year, then what would the population be in 2020?
• 3) The amount A(t) of a certain radioactive isotope after t
days is given as 𝐴 𝑡 = 𝐴𝑜 𝑒 𝑟𝑡 where 𝐴𝑜 is the initial
amount of the isotope and the rate of decay is 0.225%.
What was the initial amount if after 50 days, there was
893.6 grams left?
Small Quiz Monday:
• You will need to know for tomorrow:
• Solve exponential equations
• Compound Interest Formula
• Word Problems
• Solve natural exponential equations
• Continuously Compounded Interest
• Growth/Decay Formula
• Word Problems!
• WORD PROBLEMS!!!!!
Classwork/Homework:
• Pages 345-347 #’s 6, 8, 12, 14, 20, 22, 24, 26a, 28, 32
• This assignment will be collected!
Bell Work:
1
1
• 1) Solve: 92𝑥 ∙ ( )−5𝑥 = 27−4 ∙ ( )−1
3
• 2) Solve:
2
4𝑥
𝑒
9
= 𝑒 −15𝑥+25
• 3) The half-life of a radioactive isotope is 400 years. If there
are 600 mg of the isotope present at time t = 0, then the
amount remaining after t years is given as 𝐴 𝑡 = 600(2)−𝑡/400 .
• A) How much of the isotope is remaining after 200 years?
• B) Using the given equation, how long will it take for 75 mg of
the isotope to be left?
• C) Explain how you know the amount remaining after 2000
years without using the equation.
Examples:
• 4) Mackenzie borrows $20,000 to purchase a new car.
Her loan is to be compounded monthly over 5 years at
4.5%. What will be her monthly payments?
• 5) The local population of koala bears has been
decreasing since 1975 at a continuous rate of 6%. If the
population is currently estimated to be 100 (in 2015), what
would be the estimated population back in 1975?
1
32
• 6) Solve: 642𝑥−5 = ( )−6
Bell Work:
• 1) Colten begins a new job working for $15.25 an hour.
Every year he will get a annual 3% cost of living raise. If
he works at this same job for 30 years, what will be his
hourly wage?
LESSON 5.3
Logarithmic Functions
Lesson Essential Question:
• How are logarithmic functions related to exponential
functions and what are the different properties of
logarithms?
Logarithm
• 𝑦 = log 𝑎 𝑥 if and only if 𝑥 = 𝑎 𝑦
• x must be > 0
• y must be a real number
• How does this compare to an exponential function?
Graphs of Logarithms
• Let’s sketch the graph of a basic logarithmic function and
see how it compares to a basic exponential function.
• Sketch:
• 𝑓 𝑥 = log 2 𝑥 and 𝑔 𝑥 = 2 𝑥
Rewriting Logarithms
• We can rewrite any logarithm in exponential form.
• 𝐼𝑓 𝑦 = log 𝑎 𝑥 , 𝑡ℎ𝑒𝑛 𝑎 𝑦 = 𝑥.
• Let’s rewrite the following logarithms as exponents:
• log 2 16 = 𝑥
• log 𝑦 625 = 4
• log10 𝑚 = 3
Rewriting Exponents
• We can also rewrite exponents as logarithms.
• Ex: 𝑥 5 = 32
• Ex: (4𝑦 + 5)3 = 27
• Ex: 3 𝑥 = 2𝑥 + 1
Simplifying Logarithms
• Simplify (if possible):
• Ex:
1
log 3
81
• Ex: log 2 64
• Ex: log12 1
• Ex: log 5 −25
Bell Work:
• Simplify:
1
• 1) log 6
=?
36
• 2) log 2 16= ?
• 3) log 8 2 = ?
• 4)
1
log 32
4
=?
• 5) log 4 −16 = ?
Properties of Logarithms
• Page 350 Blue Table
• Remember These Properties!!!!!
Solving Logarithmic Equations
• Ex: log 4 5 + 𝑥 = 3
• Ex: log 5 𝑥 2 − 11 = 2
• Ex: log 3 (𝑥 − 8) = log 3 (𝑥 2 − 14)
• Ex: log 𝑥 (−5𝑥 2 + 9𝑥 + 45) = 3
Common Logarithm
• The most basic form of a logarithm:
• log 𝑥 = log10 𝑥
• If the log does not have a specified base, it is assumed to
be a base 10 log.
• Your calculator will only do base 10 logarithms!
NATURAL LOGARITHMS!!!!!!!!!
• Just like there was a natural exponential function “e”, we
also have a natural logarithmic function! YES!
• ln 𝑥 = log 𝑒 𝑥 𝑎𝑠 𝑙𝑜𝑛𝑔 𝑎𝑠 𝑥 > 0
• Just like exponential functions and logarithms are
inverses, the natural exponent and natural log are
inverses!!!
• Ex: ln 𝑥 = 2 𝑐𝑎𝑛 𝑏𝑒 𝑟𝑒𝑤𝑟𝑖𝑡𝑡𝑒𝑛 𝑎𝑠 𝑒 2 = 𝑥
Blue Table on Page 356
• These are four properties of logarithms you should know,
as well as:
• If 𝑦 = 𝑎𝑏 𝑥 , then 𝑦 = 𝑎𝑒 𝑥∙ln 𝑏
• Ex: Convert 5 ∙ 3 𝑥 to a base e expression.
Homework:
• Pages 359 – 360
• #’s 2, 4, 10, 12, 14, 16 – 32 evens
Bell Work:
• Be ready to ask questions on the homework!
• If not, begin working on the following assignment which is
due Monday! (It will be collected!)
• Pages 359 – 360 #’s 1, 3, 13 – 31 odds
Bell Work:
• Solve:
• 1) log 9 27 = 𝑥
• 2) 5𝑒 ln 𝑥 = 45
• 3) 𝑒 𝑥 ln 5 = 0.04
• 4) Solve for t. 100 = 200𝑒 −0.05𝑡
• 5) Solve for k. 40 = 160(10)1.5𝑘
Graphing Logarithms
• Lets create a table and sketch the graphs of the following:
• 𝑦 = log 3 𝑥
• 𝑦 = log 2 𝑥
• 𝑦 = ln 𝑥 (graphing calculator)
• What do you notice about these graphs compared to
exponential functions?
Shifting/Reflecting
• The graphs of logarithms behave just like any other
functions.
• Lets sketch the graphs of some functions that will shift
and/or reflect.
Class Examples:
• Pages 362 – 363 #’s 52, 54, 60, 66, 70
Homework:
• Page 360 #’s 51 - 69 odds
Bell Work:
• The current population of Bloomsburg in 2015 is
approximately 12,000. It is projected to grow continuously
in the future at a rate of 1.85%. How long will it take for
the population of Bloomsburg to double in size?
LESSON 5.4
Properties of Logarithms
Lesson Essential Question
• What are the different properties of logarithms and how
are they used when simplifying exponential and/or
logarithmic expressions?
Three More Properties of Logarithms
• ORANGE TABLE ON PAGE 364
• These properties hold true for the common logarithm and
the natural logarithm. (Blue Table Page 365)
Examples:
• Using the laws of logarithms, rewrite the expression using
logs of x, y, or z.
• Ex: log 𝑎 𝑥 5 𝑦
• Ex:
𝑥4
log 𝑎 10
𝑦
• Ex: ln
3
𝑥4
𝑦2𝑧 3
Examples:
• Using the properties of logarithms, rewrite each
expression as one logarithm.
• Ex:
1
log 𝑎
3
• Ex:
1
log 𝑎 (𝑥 2
3
𝑥 − 2 log 𝑎 𝑦
− 1) − 3 log 𝑎 𝑦 − log 𝑎 𝑧
Solving Logarithmic Equations:
• Solve each logarithmic equation. Double check to make
sure the solution you found is in fact a solution!
• Ex: log 7 (8𝑥 − 12) = log 7 2𝑥 + log 7 5
• Ex: log 4 𝑥 + log 4 (𝑥 + 4) = log 4 12
• Ex: 2 ln 𝑥 − 2 − ln 4 = ln 25
• Ex: 3 ln 𝑥 = ln 𝑒 3 + ln 8
Homework:
• Page 370 #’s 1-33 odds
Bell Work:
• Get out your homework from last night:
• Page 370 1-33 odds
• Be ready to ask questions!
Quiz Tomorrow:
• LOGARITHMS!
• (remember that to answer some logarithmic problems you
•
•
•
•
•
need to know how to change to exponential form)
On the quiz:
Solving Logarithms
Word Problems
Properties of Logarithms
WORD PROBLEMS!!!
Class Work:
• Pages 370 – 371 #’s 6, 8, 10, 12, 14, 18, 20, 22, 26, 34,
51, 52, 53, 54, 56
Bell Work:
• 1) Armaan invests $25,000 into a mutual fund that has a
continuously compounded interest rate of 4.75%. How
long will it take for Armaan to triple his investment?
• 2) Solve: ln 25 = ln 5 + ln 1 − 0.4ℎ
• 3) Solve: 2 log(𝑥 − 5) − log 2 = log 32
LESSON 5.5
Exponential and Logarithmic Equations
Lesson Essential Question
• How can we change the bases of logarithmic and
exponential functions, and how do we use the special
base formulas?
Example:
• Solve: 4 𝑥 = 20
• We can rewrite this in logarithmic form, but we still can’t
solve it. Or can we?????????
Change of Base Formula:
• log 𝑏 𝑢 =
log𝑎 𝑢
log𝑎 𝑏
• We can rewrite a logarithm with a “difficult” base as a
quotient of two common logarithms or two natural logs.
log 10
ln 10
• Ex: log 5 10 =
=
log 5
ln 5
• We can prove it!
Examples:
• Solve each equation for x first, then approximate each to
two decimal places:
• Ex: 3 𝑥 = 18
• Ex: 42𝑥−5 = 10
• Ex: 52𝑥+1 = 6 𝑥−2
Solving an Exponential Equation
• This problem is off the hook yo!!!!!!!!!!!!
• Ex: Solve for x, then approximate to the nearest 2
decimal places:
5𝑥 −5−𝑥
2
=3
Homework:
• Pages 381-382 #’s 1-31 odds only
Bell Work:
• Solve for x.
•1=
10𝑥 +10−𝑥
2
•5=
6𝑥 +6−𝑥
3
Solving a Logarithmic Equation:
• This pizzle is fo rizzle ( LOL):
• Ex: Solve: log
3
𝑥=
log 𝑥
Finding an inverse function:
• Same type of problem that we solved in the bell work.
• Example: Find the inverse function of 𝑦 =
2
𝑒 𝑥 +𝑒 −𝑥
• How could we use our graphing calculator to prove that
the functions are indeed inverses?
Example using the Logistic Curve
• A logistic curve is the graph of an equation in the form:
•𝑦=
𝑘
,
1+𝑏𝑒 −𝑐𝑥
where b, c, and k are constants, x represents
• the time, and y is the population.
• Example: Assume c = 1.1244, k = 105, and x will be the
time in days.
• A) Find the value of b if the initial population was 3.
• B) How long will it take the population to reach 90?
• C) Show that after a long period of time, the population of
this curve will become the constant k.
Class Work/Homework:
• Pages 381-383 #’s 33 – 39 odds 49, 53b, 54, 55, 56a,
56c, 57
• If you need extra practice, try the evens (2 – 40)
Unit Test
• Thursday and Friday we will be having our Unit 5 Test on
exponential and logarithmic functions.
• Here is a group of review exercises to try:
• Pages 385 – 387
• #’s 17 – 40, 45 and 46 (just find the inverse),
• 47 – 55, 58, 61, 62, 66, 67