AAT
Name:________________________________
Exponential and Logarithmic
Functions and Relations
DeGroh/Grunloh/Sokolowski
2014
1
Learning Targets
Graph an exponential functions using transformations
#1
M
DM
BU
IU
NE
BU
IU
NE
BU
IU
NE
IU
NE
IU
NE
Graph a logarithmic function using transformations
#2
M
DM
Simplify logarithmic expressions
#3
M
DM
Solve exponential equations using the same base
#4
M
DM
BU
Solve exponential equations with different bases
#5
M
DM
BU
Expand and condense logarithmic expressions using logarithmic properties
#6
M
DM
BU
IU
NE
BU
IU
NE
IU
NE
IU
NE
Solve logarithmic equations
#7
M
DM
Solve logarithmic and exponential application problems
#8
M
DM
BU
Solving compound interest application problems
#9
M
DM
BU
Video Links for Review/Enhancing Understanding
Target 1
Target 2
Target 3
Target 4
Target 5
Target 6
Target 7
Target 8/9
2
Target 1:__________________________________________________________________________________________________________________________
https://www.khanacademy.org/math/trigonometry/exponential_and_logarithmic_func/exp_growth_decay/v/graphingexponential-functions
Math in the Real World!
1. What’s happening to the home phones (landlines)?
Year
% of household
with home phones
1992
96
1998
93
2003
91
2005
90
2010
75
2011
71
2014
68
a. Does this graph model a linear function? Why or why not? _________________________________________________________
____________________________________________________________________________________________________________________________
b. Estimate the number of home phones in 2025. _________________________________
c. Will the home phones ever go away completely? ____________________________________________________________________
2. Holy Facebook! The data below represents the increase in the number of Facebook users over time.
Year
# of Facebook Users
(millions)
2004
1
2005
5.5
2006
12
2007
50
2008
100
2009
350
2010
608
2011
845
2012
1.06 billion
2013
1.2 billion
a. Describe what’s happening with Facebook’s users over
time. _____________________________________________________________
___________________________________________________________________
b. How come the graphs in #1 & #2 are different? _______
_____________________________________________________________________________
3
Ex. 1: Graph each of the functions using a table of values.
1.
y  2x
2.
X
y  ex
Y
X
In general
y  ax
In general
Y
y  ex
Domain: _____________________
Domain: _____________________
Range: _____________________
Range: _____________________
Increasing or decreasing
Increasing or decreasing
Intercept:_____________________
Intercept:_____________________
Horizontal asymptote:_________________
Horizontal asymptote:_________________
Ex. 2: Transformations of Exponential Graphs: Use #1 and #2 from above to help you sketch each graph below by hand
using transformation. Use your calculator to verify your results.
a. Graph
y  2 x 1
b. Graph
y  2 x 3
c. Graph
y  ex 1
Increasing or decreasing
Increasing or decreasing
Increasing or decreasing
Asymptote:___________
Asymptote: ____________
Asymptote: ___________
4
d. Graph
y  2x  3
e. Graph
y  e x
Intercept: ___________
Intercept: _____________
Increasing or decreasing
Increasing or decreasing
Asymptote:___________
Asymptote: ____________
f. In general
y  a x c  b describe the transformations of each (Shifts/ Reflections).
If c is positive, then ______________________________________________________
If c is negative, then _____________________________________________________
If b is positive, then _____________________________________________________
If b is negative, then _____________________________________________________
If there is a negative sign in front of x, then ___________________________________
Ex 3: Between 1990 and 1999 college tuition increased at a rate of 7.2% per year. The cost of tuition, C (in thousands),
can be modeled by the equation C  15 1.072  where t is the number of years since 1990. If the cost of tuition
t
continued to grow at the same rate, what would the cost of the tuition be today in 2014?
5
Target 1 Practice:
Sketch the graphs using transformations.
1. Graph
y  4x
2. Graph
y  4 x
3. Graph
y  e x2
Intercept: ___________
Intercept: _____________
Intercept: _____________
Asymptote:___________
Asymptote: ____________
Asymptote: ___________
Increasing or decreasing
Increasing or decreasing
Increasing or decreasing
4. Graph
y  4x  3
5. Graph
y  e x 1
6. Graph
y  4 x  1
Intercept: ___________
Intercept: _____________
Intercept: _____________
Asymptote:___________
Asymptote: ____________
Asymptote: ___________
Increasing or decreasing
Increasing or decreasing
Increasing or decreasing
7. From 1990 to 1997, the number of cell phone subscribers, S (in thousands), in the US can be modeled by the equation
S  5535.33 1.413 where t is the number of years since 1990. Find the number of cell phone subscribers in the year
t
2010.
6
Target 2:__________________________________________________________________________________________________________________________
https://www.khanacademy.org/math/algebra/logarithms-tutorial/logarithm_basics/v/graphing-logarithmic-functions
Math in the Real World!
#thetrendoftwitter
1. What’s happening to twitter?
Year
# of people signed
up for Twitter
(millions
2008
0
2009
18
2010
64
2011
137
2012
185
2013
218
a. Does this graph model a linear function? Why or why not? _________________________________________________________
____________________________________________________________________________________________________________________________
b. What is a factor that could cause such a growth in the number of twitter accounts in the beginning?________________
____________________________________________________________________________________________________________________________________
c. Will the number of twitter accounts ever stop increasing? _______________________________________________________________
_________________________________________________________________________________________________________________________________
7
Ex. 1: A logarithmic function is an inverse function of an exponential function. Make a table of values for the
exponential function. Interchange the coordinates to graph the log function. Make a table of values for the log function
also.
1.
y  2x
2.
X
y  log 2 x rewrite as _________________
Y
X
In general
y  ax
In general
Y
y  log a x
Domain: _____________________
Domain: _____________________
Range: _____________________
Range: _____________________
Increasing or decreasing
Increasing or decreasing
Intercept:_____________________
Intercept:_____________________
Asymptote:_________________
Asymptote:_________________
3. y  ln x rewrite as _________________ and then as __________________
In general
X
y  ln x
Y
Domain: _____________________
Range: _____________________
Increasing or decreasing
Intercept:_____________________
Asymptote:_________________
8
Ex 2: Transformations of Logarithmic graphs: Sketch each graph by hand then use your graphing calculators to check.
a. Graph f ( x)  log10 x
b. Graph f ( x)   log10 x
c. Graph f ( x)  log10 ( x)
Describe Transformation:
Describe Transformation:
Describe Transformation:
____________________
____________________
______________________
Asymptote:_____________
Asymptote:___________
Asymptote:_______________
d. Graph f ( x)  log10 ( x  3)
e. Graph f ( x)  log10 x  4
f. Graph f ( x)  log10 ( x  2)  3
Describe Transformation:
Describe Transformation:
Describe Transformation:
____________________
____________________
______________________
Asymptote:_____________
Asymptote:___________
Asymptote:_______________
g. Graph f ( x)  ln( x  1)
h. Graph f ( x)  ln x  2
i. Graph f ( x)   ln( x)  2
Describe Transformation:
Describe Transformation:
Describe Transformation:
____________________
____________________
______________________
Asymptote:_____________
Asymptote:___________
Asymptote:_______________
9
Target 2 Practice
Sketch graphs by hand using transformations, then use your graphing calculators to check.
1. Graph f ( x)  log 6 ( x)
2. Graph f ( x)  log 6 x  1
3. Graph f ( x)  log 6 ( x  2)
Describe Transformation:
Describe Transformation:
Describe Transformation:
____________________
____________________
______________________
Asymptote:_____________
Asymptote:___________
Asymptote:_______________
4. Graph f ( x)  ln( x)  2
5. Graph f ( x)  ln x  3
6. Graph f ( x)  ln( x  3)  1
Describe Transformation:
Describe Transformation:
Describe Transformation:
____________________
____________________
______________________
Asymptote:_____________
Asymptote:___________
Asymptote:_______________
7. The magnitude of an earthquake can be measured by the formula: M  log
x
where x is the intensity
x0
measured in millimeters and x0 is the distance from the epicenter. Find the magnitude of an earthquake that
measures 7,943 millimeters and registered 100 kilometers from the center.
10
Target 3:_________________________________________________________________________________________________________________________
1. https://www.khanacademy.org/math/algebra/logarithms-tutorial/logarithm_basics/v/logarithms
2. https://www.khanacademy.org/math/algebra/logarithms-tutorial/logarithm_basics/v/fancier-logarithm-expressions
Definition of a Logarithmic Function:
The Natural Logarithmic Function:
Ex. 1: Simplify
1. log 2 8
2. log3 9
3. log5 625
4. log 4 64
5. log 6 1
6. log 9 81
7. log8 512
8. log1000
Ex. 2: Simplify
1. log 4
1
16
2. log 5
1
125
3. log 7
1
49
4. log 0.10
11
Ex. 3: Simplify
1. ln1
2. ln e
Example 4:
3. ln e 2
4. ln 3
Use a calculator to evaluate the function at the indicated value of x. Round your
result to three decimal places. To graph log base 10 use LOG button on calculator.
To graph log base e use LN button on calculator.
4
5
a.
f ( x)  log10 x ,
x  2.5
b. f ( x)  log10 x , x 
c.
f ( x)  log10 x ,
x  2
d.
f ( x)  1.9log10 x , x  4.3
e.
f ( x)  ln x ,
x  18.31
f.
f ( x)  3ln x , x  0.75
12
Target 3 Practice:
Mixed Practice
1
27
3. log1
4. log 6 36
1
5 25
7. log1 1
8. ln 5
11.
12. ln e
1. log 2 16
2. log 3
5. ln e3
6. log
9.
10. log 2 32
log7 343
log 9 1
Use your calculator as an aid. Round all answers to the thousandths place.
13.
f ( x)  log10 x ,
15.
f ( x)  ln x ,
x5
x
5
4
14. f ( x)  6log10 x , x  2.3
16.
f ( x)  3ln x , x  12.1
13
Quiz Review #1 (Targets 1-3)
Target 1/2: Sketch the exponential or logarithmic by hand. Then describe the transformation and state the new
placement of asymptote.
1. Graph f ( x)  log3 ( x)
2. Graph f ( x)  log 4 x  1
3. Graph f ( x)  e( x  2)
Describe Transformation:
Describe Transformation:
Describe Transformation:
____________________
____________________
______________________
Asymptote:_____________
Asymptote:___________
Asymptote:_______________
4. Graph f ( x)  ln( x  1)  3
5. Graph f ( x)  4 x  3
6. Graph f ( x)  e( x 3)  1
Describe Transformation:
Describe Transformation:
Describe Transformation:
____________________
____________________
______________________
Asymptote:_____________
Asymptote:___________
Asymptote:_______________
Target 3: Simplify each logarithmic expression.
7. log4 64
8. log6
1
216
9. log5 125
10. lne 5
11. 2e ln 4
14
Target 4:__________________________________________________________________________________________________________________________
1. http://www.brightstorm.com/math/algebra-2/inverse-exponential-and-logarithmic-functions/solving-exponentialequations-with-the-same-base-problem-1/
2. http://www.brightstorm.com/math/algebra-2/inverse-exponential-and-logarithmic-functions/solving-exponentialequations-with-the-same-base-problem-2/
3. http://www.brightstorm.com/math/algebra-2/inverse-exponential-and-logarithmic-functions/solvingexponential-equations-with-the-same-base-problem-3/
Example 1: Solve the exponential equation.
1. 3x  35
2. 24 x  212
5. 24 x  4
6. 72  2  65 x  2
3. 5 x 1  52 x 5
7. 43 x 
4.
1
4
4x  42 x  4( x8)
8. 34( x 3)  93 x
Target 5:__________________________________________________________________________________________________________________________
https://www.khanacademy.org/math/trigonometry/exponential_and_logarithmic_func/exponential-modeling/v/solveexponentials
Example 2: Solve exponential equations with different bases
1. 3  2 x
2. 4x8  12
3. e3 x  2  7
4.
10  5e4 x
(same as before but
use new method)
5. 2 x  7  10
6. 6  25 x 2  2
7. 43 x 
1
4
8. Why does 3x 2  2
Have no solution?
15
Targets 4-5 Practice
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
16
Target 6:__________________________________________________________________________________________________________________________
1. https://www.khanacademy.org/math/algebra/logarithms-tutorial/logarithm_properties/v/introduction-to-logarithmproperties
2. https://www.khanacademy.org/math/algebra/logarithms-tutorial/logarithm_properties/v/introduction-to-logarithmproperties--part-2
3. http://www.brightstorm.com/math/precalculus/exponential-and-logarithmic-functions/condensing-logarithmsproblem-3/
Warm ups:
Solve
1. 3x  9
2. 23 x  3
3. e3 x 1  2  7
Example 1: Logarithmic properties
Note: log a b c  c log a b  Exponents can be brought out front
a. log 3 32 
b. log 2 4 x3 =
c. log 3 27 2 x =
d. log 3
1

9
Note: Multiplication log a (mn)  loga m  loga n  addition same as exponents
e. log 2 48
f. log a x 2 y 3
g. log3 27xq
h. ln ex 2
Note: Division log b
i. log 2
m
 logb m  logb n  Subtraction same as exponents too.
n
16

x
9 x3
k. log 3
y
j. logb
x2

y3
ex5
l. ln 2
yz
17
Example 2: Using logarithmic properties condense the logarithms. (Make one single logarithm)
1
log 2 4
2
a. log3 5  log3 7
b. 2 log 2 3 
c. 3log 4 3  5log 4 x
d. 3log a x  4log a y  7 loga m
Practice: Target 6
Directions: Expand each logarithm.
1.
2.
3.
4.
5.
x2 y5
ln 4 =
wz
6.
log100x5 y 2 
Directions: Condense each logarithm.
7.
8.
9.
10.
11.
3log 5 x  2log 5 y  4log 5 z =
12.
4log 2 x  3log 2 z  2log 2 m =
18
Target 7:__________________________________________________________________________________________________________________________
1. https://www.khanacademy.org/math/algebra/logarithms-tutorial/logarithm_properties/v/solving-logarithmicequations
2. http://www.brightstorm.com/math/precalculus/exponential-and-logarithmic-functions/solving-a-logarithmicequation/
Warm Ups:
Condense:
1. log3 x  2log3 y
2. 3ln y  5ln z
3. log x  3log y  4 log z
Expand:
 3xy 4 
4. ln(3x )
5. log4  5 
 z 
Example 1: Solve each logarithmic equation .
a. log3 5 x  3
b. ln( x  3)  2
log 2 ( x  3)  log 2 5  3
d. 7  3log 3 ( x  2)  16
e.
f. log 2 ( x  3)  log 2 (2 x  1)
g. log3 (3x  7)  log 2 ( x  4)


h. log 5 p 2  2  log 5 p
c. log 4 (2 x  1)  1
i. log 2 x  log 2 ( x  2)  3
19
Practice target 7:
1.
2.
3.
4.
5.
6.
7.
8.
9. log3 ( x  6)  log3 x  3
20
Quiz Review #2 (Targets 4-7)
Target 4: Solve exponential equations using the same bases.
1. 43  42 x5
3 x 6

3. 5
x4
x
2. 3  9
1
5x
Target 5: Solve exponential equations using different bases.
x 2
4. 2  3
2 x 3
 4  10
6. 3e
( x 4)
5. 9  3 2
Target 6: Expand/Condense logarithmic equations using logarithmic properties.
Expand
4
7. log 3 9x
Condense
10. 3log x  5log y
8. ln
2 x5
y4
9. log(10 x5 )
11. 7 ln x  3ln y  2 ln z
12. log 2 x  log 2 y  log 2 z
14. 4 ln( x  5)  12
15. 4log3 ( x  4)  7  11
Target7: Solve a logarithmic equation
13. 2 log x  4
16. log( x  3)  log(5 x  11)
17. log3 ( x)  log3 ( x  6)  2
21
Target 8:__________________________________________________________________________________________________________________________
1. http://www.youtube.com/watch?v=7yoru7s2wrs
2. http://www.youtube.com/watch?v=Lj9qNmLRmJ8
General Exponential Growth Formula:
General Exponential Decay Formula:
Example 1: In 1985, there were 285 cell phone subscribers in the small town of Centerville. The number of
subscribers increased by 75% per year after 1985. How many cell phone subscribers were in Centerville in 1994?
Example 2: An adult takes 400 mg of ibuprofen. Each hour, the amount of ibuprofen in the person’s system
decreases by about 29%. How much ibuprofen is left after 6 hours?
Example 3: The foundation of your house has about 1,200 termites. The termites grow at a rate of about 2.4% per
day. How long until the number of termites doubles?
Example 4: You drink a beverage with 120 mg of caffeine. Each hour, the caffeine in your system decreases by
about 12%. How long until you have 10mg of caffeine?
22
Example 5: Each year the local country club sponsors a tennis tournament. Play starts with 128 participants.
During each round, half of the players are eliminated. How many players remain after 5 rounds?
Example 6: Desalination is the process of producing fresh water from salt water. How much fresh water can be
produced after 10 hours from a desalination process using the formula: y  18.27  31.03ln t where y is the
amount of fresh water to be produced after t hours.
Example 7: On a college campus of 5000 students, one student returned from vacation with a contagious flu virus.
The spread of the virus is modeled by y 
5000
, t0
1  4999e 0.8t
Where y is the total number infected after t days. The college will cancel classes when 40% or more of the students
are infected.
a. How many students were infected after 5 days?
b. After how many days will the college cancel classes?
23
Target 8 Practice:
1. You have inherited land that was purchased for $30,000 in 1960. The value of the land increased by
approximately 5% per year. What is the approximate value of the land in the year 2011?
2. You buy a new computer for $2100. The computer decreases by 50% annually. When will the computer
have a value of $600?
3. A new car that sells for $18,000 depreciates 25% each year. Write a function that models the value of the car.
Find the value of the car after 4 years.
4. The population of Winnemucca, Nevada, can be modeled by P=6191(1.04)t
where t is the number of years since 1990. What was the population in 1990? By what percent did the population
increase by each year?
5. Knowledge retention of a student who works independently is modeled by x (t )  91  30log(t  1) . Knowledge
retention of students who work in groups in modeled by y (t )  88  15log(t  1) . Which method of learning do
students retains the most knowledge in after 2 months?
6. The growth of height of trees can be described by
h
120
. Suppose the height h, in feet, of a tree at
1  200e 0.2t
age t, in years, is given by the equation above. Find the age of the tree when the height is 50 feet.
24
Target 9:__________________________________________________________________________________________________________________________
http://www.brightstorm.com/math/precalculus/exponential-and-logarithmic-functions/compound-interest-finitenumber-of-calculations-problem-1/
Formulas for compound interest
For n compounding per year:
A =___________________________
For continuous exponential growth/decay.
P = ___________________________
r =_____________________________________________
t = _________________________
n =_____________________________________
Ex 1) Find the balance in the account after $1000 is deposited into an account that earns 3% annual interest
compounded monthly for 5 years.
Ex 2) A total of $12,000 is invested at an annual interest rate of 4.5%. Find the balance after 4 years if the interest
is compounded quarterly.
Ex 3) If Melinda invests her $80,000 winnings from Publishers Clearing House at 9% compounded monthly, then
what will the investment be worth at the end of 20 years? How much interest will be earned during the 20 years?
Ex 4) If a credit union pays 6.5% annual interest compounded daily, then what will a deposit of $2300 be worth
after 5 years and 3 months?
25
Ex 5)To attract funds, the financially troubled Commercial Federal Savings and Loan offered 9¾% annual interest
compounded daily on certificates of deposit. At this rate, how much interest would a deposit of $30,000 earn in 18
months?
Ex 6) The half-life of a radioactive substance is the time it takes for half of the atoms of the substance to become
disintegrated. All life on Earth contains the radioactive element Carbon-14, which decays continuously at a fixed rate.
The half-life of Carbon-14 is 5760 years. That is, every 5760 years half of a mass of Carbon-14 decays away.
A. What is the value of “k” for Carbon-14?
Ex 7) Rebecca is examining the bones of a chupacabra and estimates that they contain only 3% as much Carbon-14 as
they would have contained when the animal was alive. How long ago did the chupacabra die?
Ex 8) You spill 20 tons of deGrohnium which has a half life of 2300 years. How long will it be until there is only 3 tons
left?
26
Test Review
Target 1: Sketch the exponential graph using transformations.
1.
y  3x
2.
y  3x  2
3. y  3x  1
Increasing or Decreasing
Increasing or decreasing
Increasing or decreasing
Asymptote:__________________
Asymptote:___________________
Asymptote:____________________
Target 2: Sketch the logarithmic function using transformations.
y  log 4 x
5. y  log 4 ( x  1)
6.
Describe transformation:
Describe transformation:
Describe transformation:
_______________________________
_______________________________
________________________________
Asymptote:____________________
Asymptote:___________________
Asymptote:_____________________
4.
y  log 4 ( x  2)  1
Target 3: Simplify
1
27
7. log 2 16
8. log8 64
9. log 3
11. ln e
12. ln e 3
13. log100
10. log 7 1
14. log
1
10
27
Target 4: Solve exponential equations with the same base.
15. 52 x  57
16. 4 x  64
18. 80  8  103 x1
19. 25 x 
1
4
17. 3x  33 x1  312
20. 64 x3  363 x
Target 5: Solve exponential equations with different bases. Round to the nearest thousandth.
21. 3x  14
22.
5x2  7
24. 4 x  3  11
25. 123( x1)  9
23. e5 x  10
26. 5  2e 2 x  14
Target 6: Expand or condense each logarithm using logarithmic properties.
Expand
27. log 2 3x 4
28. ln
xy
a2
29.
log 100 y 7 
28
Condense
30.
4log3 x  log3 y
31. ln a  ln b  3ln c
32. 4log x  6log y  log z
Target 7: Solve a logarithmic equation.
33.
log 4 x  7
34. 2  log 2 ( x  4)  12
35.
36.
log 2 ( x  10)  log 2 (3x  2)
37. log3 ( x)  log3 ( x  8)  2
38. ln( x  5)  ln( x)  ln 5
5ln x 1  11
Target 8: Solve the exponential or logarithmic application problem.
39. The bear population decreases at a rate of 2% per year. There are 1573 bear this year. Write a function that
models the bear population growth. How many bear will there be in 10 years?
40. The number of followers to GNHS nice things Twitter page increases at a rate of 75% per year. The page
started out with 100 followers. How many years did it take for it to have 542 followers?
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41. In a typing class, the average number of words per minute, N, after t week of lessons was found to be modeled
by N 
157
. Find the number of weeks necessary to type 50 words.
1  5.4e 0.12 t
Target 9: Solve compound interest application problems.
42. Find the balance in an account that had an initial deposit of $2350 that was compounded monthly at an
interest rate of 6.5% for 8 years.
43. You are disarming a bomb which will go off once the temperature of the trigger reaches 100 degrees. If the
temperature is following exponential growth and you initially read 45 degrees and then ½ hour later read 80 degrees,
how long do you have to diffuse the bomb?
44. A you are working on a murder case where the killer used a radioactive poison. If the half-life of the poison is 5
hours and 12% of the poison is remaining, how long ago was the person poisoned?
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