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4.1 Exponential Functions and Their Graphs
In this section you will learn to:
- evaluate exponential functions
- graph exponential functions
- use transformations to graph exponential functions
- use compound interest formulas
An exponential function f with base b is defined by
f ( x )= b^ x or y = bx , where b > 0, b ≠ 1, and x is any real number.
Note: Any transformation of y = bx is also an exponential function.
Example 1: Determine which functions are exponential functions. For those that are not, explain why they are not exponential functions.
(a) f ( x )= 2 x + 7 Yes No __________________________________________________
(b) g ( x )= x^2 Yes No __________________________________________________
(c) h ( x )= 1 x Yes No ___________________________________________________
(d) f ( x )= xx Yes No ___________________________________________________
(e) h ( x )= 3 ⋅ 10 − x Yes No __________________________________________________
(f) f ( x )= − 3 x +^1 + 5 Yes No __________________________________________________
(g) g ( x )= (− 3 ) x +^1 + 5 Yes No __________________________________________________
(h) h ( x )= 2 x − 1 Yes No __________________________________________________
Example 2: Graph each of the following and find the domain and range for each function.
(a) f ( x )= 2 x domain: __________
range: __________
(b)
x g x
( ) domain: __________
range: __________
Characteristics of Exponential Functions
x f ( x )= b
b > 1 0 < b < 1
Domain: Range:
Transformations of g ( x ) = b
x
( c > 0): (Order of transformations is H S R V.)
Horizontal: g ( x ) = bx + c (graph moves c units left)
g ( x ) = bx −^ c (graph moves c units right)
Stretch/Shrink: g ( x )= cbx (graph stretches if c > 1)
(Vertical) (graph shrinks if 0 < c < 1)
Stretch/Shrink: g ( x )= bcx (graph shrinks if c > 1)
(Horizontal) (graph stretches if 0 < c < 1)
Reflection: g ( x )=− bx (graph reflects over the x -axis)
g ( x ) = b −^ x (graph reflects over the y -axis)
Vertical: g ( x )= bx^ + c (graph moves up c units)
g ( x )= bx^ − c (graph moves down c units)
Periodic Interest Formula Continuous Interest Formula
nt
n
r A P
= 1 + A = Pert
A = balance in the account (Amount after t years)
P = principal (beginning amount in the account)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = time (in years)
Example 5: Find the accumulated value of a $5000 investment which is invested for 8 years at an interest rate of 12% compounded:
(a) annually
(b) semi-annually
(c) quarterly
(d) monthly
(e) continuously
4.1 Homework Problems
- Use a calculator to find each value to four decimal places.
(a) 5 3 (b) 7 π (c) 2 −^5.^3 (d) e^2 (e) e −^2 (f) − e^0.^25 (g) π−^1
2. Simplify each expression without using a calculator. (Recall: b n^ ⋅ bm = bn + m and ( ) mn
m n b = b )
(a) 6 2 6 2 (b) (^ )^
2 2
3 (c) (^ )^
2 8
b (d) (^ )^
3 3 5 (e) 2
1 2
1 4 4 (f) b^12 b^3
For Problems 3 – 14, graph each exponential function. State the domain and range for each along with the equation of any asymptotes. Check your graph using a graphing calculator.
- f ( x )= 3 x 4. f ( x )= −( 3 x ) 5. f ( x ) = 3 − x 6.
x f x
- f ( x )= 2 x − 3 8. f ( x ) = 2 x −^3 9. f ( x )= 2 x +^5 − 5 10. f ( x )= − 2 − x
- f ( x )= − 2 x +^3 + 1 12. 4 2
3 ^ −
x − f x 13. f ( x )= e −^ x + 2 14. f ( x )= − ex +^2
- $10,000 is invested for 5 years at an interest rate of 5.5%. Find the accumulated value if the money is (a) compounded semiannually; (b) compounded quarterly; (c) compounded monthly; (d) compounded continuously.
- Sam won $150,000 in the Michigan lottery and decides to invest the money for retirement in 20 years. Find the accumulated value for Sam’s retirement for each of his options: (a) a certificate of deposit paying 5.4% compounded yearly (b) a money market certificate paying 5.35% compounded semiannually (c) a bank account paying 5.25% compounded quarterly (d) a bond issue paying 5.2% compounded daily (e) a saving account paying 5.19% compounded continuously
4.1 Homework Answers: 1. (a) 16.2425; (b) 451.8079; (c) .0254; (d) 7.3891; (e) .1353; (f) -1.2840;
(g) .3183 2. (a) 36 2 ; (b) 9; (c) b^4 ; (d) 125; (e) 4; (f) b^33 3. Domain: ( −∞, ∞); Range: ( 0 ,∞) ;
y = 0 4. Domain: ( −∞, ∞); Range: (−∞, 0 ); y = 0 5. Domain: ( −∞, ∞); Range: ( 0 ,∞ ); y = 0
6. Domain: ( −∞, ∞); Range: ( 0 ,∞) ; y = 0 7. Domain: ( −∞, ∞); Range: ( − 3 ,∞); y =− 3 8. Domain: ( −∞, ∞); Range: ( 0 ,∞) ; y = 0 9. Domain: ( −∞, ∞); Range: ( − 5 ,∞); y =− 5 10. Domain: ( −∞, ∞); Range: (−∞, 0 ); y = 0 11. Domain: ( −∞, ∞); Range: (−∞ , 1 ); y = 1 12. Domain: ( −∞, ∞); Range: ( − 4 ,∞); y =− 4 13. Domain: ( −∞, ∞); Range: ( 2 ,∞ ); y = 2 14. Domain: ( −∞, ∞); Range: (−∞, 0 ); y = 0 15. (a) $13,116.51; (b) $13,140.67; (c) $13,157.04;
(d) $13,165.31 16. (a) $429,440.97; (b) $431,200.96; (c) $425,729.59; (d) $424,351.12;
(e) $423,534.
Example 4: The exponential function f ( x )= 84. 5 ( 1. 012 ) x models the population of Mexico, f ( x ), in
millions, x years after 1986.
(a) Without using a calculator, substitute 0 for x and find Mexico’s population in 1986.
(b) Estimate Mexico’s population, to the nearest million in the year 2000.
(c) Estimate Mexico’s population, to the nearest million, this year.
Example 5: College students study a large volume of information. Unfortunately, people do not
retain information for very long. The function f ( x )= 80 e −^0.^5 x + 20 describes the percentage of
information, f ( x ), that a particular person remembers x weeks after learning the information (without
repetition).
(a) Substitute 0 for x and find the percentage of information remembered at the moment it is first learned.
(b) What percentage of information is retained after 1 week? ______ 4 weeks? _______ 1 year? _______
Radioactive Decay Formula:
The amount A of radioactive material present at time t is given by h
t
A A
−
= 0 ( 2 ) where A 0 is the amount
that was present initially (at t = 0) and h is the material’s half-life.
Example 6: The half-life of radioactive carbon-14 is 5700 years. How much of an initial sample will remain after 3000 years?
Example 7: The half-life of Arsenic-74 is 17.5 days. If 4 grams of Arsenic-74 are present in a body initially, how many grams are presents 90 days later?
Logistic Growth Models: Logistic growth models situations when there are factors that limit the
ability to grow or spread. From population growth to the spread of disease, nothing on earth can exhibit exponential growth indefinitely. Eventually this growth levels off and approaches a maximum level (which can be represented by a horizontal asymptote).
Logistic growth models are used in the study of conservation biology, learning curves, spread of an epidemic or disease, carrying capacity, etc. The mathematical model for limited logistic growth is given
by: (^) bt bt
ae
c
or A
ae
c
f t − −
( ) , where a, b, and c are constants, c > 0 and b > 0.
As time increases ( t →∞), the expression ae − bt →_______ and A →_______.
Therefore y = c is a horizontal asymptote for the graph of the function. Thus c represents the limiting size.
Example 8: The function (^) t e
f t 0. 06 1 1999
= describes the number of people, f ( t ),who have
become ill with influenza t weeks after its initial outbreak in a town with 200,000 inhabitants.
(a) How many people became ill with the flu when the epidemic began? __________
(b) How many people were ill by the end of the 4th^ week? __________
(c) What is the limiting size of f ( t ), the population that becomes ill? __________
(d) What is the horizontal asymptote for this function? __________
Example 9: The function (^) t e
f t 0. 2 1
= is a model for describing the proportion of correct responses,
f ( t ), after t learning trials.
(a) Find the proportion of correct responses prior to learning trials taking place. __________
(b) Find the proportion of correct responses after 10 learning trials. __________
(c) What is the limiting size of f ( t )as continued trials take place? __________
(d) What is the horizontal asymptote for this function? __________
(e) Sketch a graph of this function.
- The 1986 explosion at the Chernobyl nuclear power plant in the former Soviet Union sent about 1000
kilograms of radioactive cesium-137 into the atmosphere. The function ( ) 1000 ( 0. 5 )^30
x f x = describes the amount, f ( x ), in kilograms, of cesium-137 remaining in Chernobyl x years after 1986. If even 100 kilograms of cesium-137 remain in Chernobyl’s atmosphere, the area is considered unsafe for human habitation. Find f ( 60 )and determine if Chernobyl will be safe for human habitation by 2046.
- The logistic growth function (^) t e
f t −
( ) describes the number of people, f ( t ), who have
become ill with influenza t weeks after its initial outbreak in a particular community.
(a) How many people became ill with the flu when the epidemic began?
(b) How many people were ill by the end of the fifth week?
(c) What is the limiting size of the population that becomes ill?
- The logistic growth function (^) x e
P x 0. 122 1 271
= models the percentage, P ( x )of Americans who
are x years old with some coronary heart disease.
(a) What percentage of 20-year-olds have some coronary heart disease?
(b) What percentage of 80-year-olds have some coronary heart disease?
4.2 Homework Answers: 1. (a) g ( x ) = 2 x +^2 ; (b) g ( x )= − 2 x +^2 ; (c) g ( x )= − 2 x +^2 + 3 2. (a) g ( x )=− ex ;
(b) g ( x ) = e − x ; (c) g ( x )= 4 − ex 3. about 322.7 million 4. (a) about 1732; (b) 3000;
(c) about 5196; (d) 9000 5. (a) about $35,917.13; (b) $5.30 6. 250; no 7. (a) about 20 people;
(b) about 2883 people; (c) 100,000 people 8. (a) about 3.7%; (b) about 88.6%
4.3 Logarithmic Functions and Their Graphs
In this section you will learn to:
- change logarithmic form ↔ exponential form
- evaluate natural and common logarithms
- use basic logarithmic properties
- graph logarithmic functions
- use transformations to graph logarithmic functions
The logarithmic function with base b is the function f ( x )= log bx. For x > 0 and b > 0, b ≠ 1,
y = log b x is equivalent to b y^ = x.
Example 1: Complete the table below:
Logarithmic Form Exponential Form Answer
log 10100 = x
3 =log 7 x
2 =log b 25
log 28 = a
log 1010
log e e = x
log 273
6 x =
b^0 = 1
23 = x
e^1 = x
b^2 = 36
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f ( x )= b^ x
and f −^1 ( x )=log bx are inverse functions
of each other. If f ( x )= 2 x , then f −^1 =_________
If f ( x )= 10 x , then f −^1 =_________
If f ( x )= ex , then f −^1 =_________
Example 4: Graph f ( x )= 2 x and g ( x )= log 2 x.
Characteristics of Inverse Functions:
y = b
x
y = log bx
Domain: Range: Domain: Range:
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Example 5: Use f ( x )= log 2 x to obtain the graph g ( x )= log 2 ( x + 3 )+ 4. Also find the domain, range,
and the equation of any asymptotes of g.
Domain: ________________
Range: _________________
Asymptote(s): _____________
Example 6: Use f ( x )= log 3 x to obtain the graph g ( x )= log 3 ( 4 − x ). Also find the domain, range, and the
equation of any asymptotes.
Domain: ________________
Range: _________________
Asymptote(s): _____________
Example 9: Let f ( x )= 2 x +^5 − 16
(a) Find the domain and range of f.
(b) Find the equation of the asymptote for the graph of f.
(c) Evaluate f (− 3 ).
(d) Find the x and y- intercepts of f.
(e) Find an equation for the inverse of f.
(f) Find the domain and range of the inverse.
Example 10: Let f ( x )= log 3 ( x + 9 )− 4
(a) Find the domain and range of f.
(b) Find the equation of the asymptote for the graph of f.
(c) Evaluate f ( 18 ).
(d) Find the x and y- intercepts of f.
(e) Find an equation for the inverse of f.
(f) Find the domain and range of the inverse.
4.3 Homework Problems
Write each equation in logarithmic form.
- 53 = 125 2. 9
3 −^2 = 3. 49 = 7 4. mn^ = p 5. 27 3
3 ^ =
−
Write each equation in exponential form.
- log 4 64 = 3 7. log 10 1000 = 3 8. log (^) π π= 1 9. log 5 1 = 0 10. 3 8
log 2 =−
Find each value of x without using a calculator.
- log 8 64 = x 12. log 64 8 = x 13. log 2 8 = x 14. log 8 2 = x 15. = x 8
log 2
1
- log 8 = x 2
1 17.^ log 9 x^ =^2 18.^ log 5 x^ =^1 19.^ log^ x^7 =^1 20.^ log 8 x =^0
21. 3 log 35 = x 22. π log^ π x = 7 23. log 3 3 = x 24. = x
log 9 25. x log^8 2 = 2
- ln e^2 = x 27. ln x = 1 28. ln x = 0 29. ln( x − 2 )= 0 30. ln x =− 1
- Graph f ( x )= 2 x + 1 and g ( x )= log 2 ( x − 1 )on the same graph. Find the domain and range of each and
then determine whether f and g are inverse functions.
For problems 32 - 35, use the graph of f ( x )= log 3 x and transformations of f to find the domain, range, and
asymptotes of g.
- g ( x )= log 3 ( x + 3 ) 33. g ( x )= 3 +log 3 x 34. g ( x )= −log 3 (− x ) 35. log ( 5 ) 2
g ( x )= − 3 x −
For problems 36 - 39, use the graph of f ( x )= ln x and transformations of f to find the domain, range, and asymptotes of g.
- g ( x )= 3 ln x 37. g ( x )= ln 3 x 38. g ( x )= 5 +ln( x + 2 ) 39. g ( x )=ln( 5 − x )
Use a calculator to find each value to four decimal places.
- log 10 17 41. log 3. 5 42. log e 5 43. ln 63 44. (log 2 )(ln 2 )
Evaluate or simplify each expression without using a calculator.
- log 1000 46. 1000
log 47. log 10 48. 10 log 7 49. log 1
- ln 1 51. ln e 52. ln e^7 53. (^7)
ln e
- ln ex
- log 3 (log 28 ) 56. log 3 (log 3 (log 327 )) 57. ln(log 4 (log 216 )) 58. log(ln e )
4.4 Applications of Logarithmic Functions
In this section you will learn to:
- use logarithms to solve geology problems
- use logarithms to solve charging battery problems
- use logarithms to solve population growth problems
Richter Scale
If R is the intensity of an earthquake, A is the amplitude (measured in micrometers), and P is the period of time (the time of one oscillation of the Earth’s surface, measured in seconds), then
P
A
R =log
Charging Batteries
If M is the theoretical maximum charge that a battery can hold and k is a positive constant that depends on the battery and the charger, the length of time t (in minutes) required to charge the battery to a given level C is given by
M
C
k
t ln 1
Population Doubling
Time
If r is the annual growth rate and t is the time (in years) required for a population to double, then
r
t
ln 2
Example 1: Find the intensity of an earthquake with amplitude of 4000 micrometers and a period of 0.07 second.
Example 2: An earthquake has a period of ¼ second and an amplitude of 6 cm. Find its measure on the Richter scale. (Hint: 1 cm = 10,000 micrometers.)
Example 3: How long will it take to bring a fully discharged battery to 80% of full charge? Assume that k = 0. 025 and that time is measured in minutes.
Example 4: The population of the Earth is growing at the approximate rate of 1.7% per year. If this rate continues, how long will it take the population to double?
4.5 Properties of Logarithms
In this section you will learn to:
- use the product, quotient, and power rules
- expand and condense logarithmic expressions
- use the change-of-base property
Properties of Exponents
n
n b
b
− (^) = b (^0) = 1 b n (^) ⋅ bm = bn + m ( ) mn m n b = b
m n n
m b b
b (^) − = ( ab ) n^ = anbn n
n n
b
a b
a =
Logarithmic Properties
Involving One log^ b b =^1 log^ b^1 =^0
Inverse Properties (^) log (^) b b x = x b log b^ x = x
Product Rule log^ b ( MN^ )=log bM +log bN
Quotient Rule N M N
M
log b =log b −log b
Power Rule M^ p bM
p
log b = log
M, N, and b are positive real numbers with b ≠ 1.
Example 1: Use the product rule to expand the logarithmic expressions. log^ b ( MN^ )=log bM +log bN
(a) log 3 x (b) log 1000 x
(c) log 1000 = (d) ln ( x^2 + 2 x )
(e) ln ex (f) ln 3 xy ( z + 1)
