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71 AMHS Precalculus - Unit 5
Unit 5: Exponential and Logarithmic Functions
Rational exponents
If b is a real number and n and m are positive and have no common factors, then n
mb = ( )m n nmb b
Laws of exponents
a)
b)
c)
d)
e)
f)
g)
Ex. 1: Simplify
a) 1
38
( )27
b) 5
29
Exponential Function
If 0b and 1b , then an exponential function ( )y f x is a function of the form ( ) xf x b .
The number b is called the base and x is called the exponent.
72 AMHS Precalculus - Unit 5
Ex. 2: Graph each of the given functions
a) ( ) 2xf x and ( ) 4xf x
b) 1
( ) 2 ( )2
x xf x and 1
( ) 4 ( )4
x xf x
73 AMHS Precalculus - Unit 5
In general …
( ) xf x b
Domain: Range: Intercept: Horizontal Asymptote:
1( ) ( )x xf x b
b
Domain: Range: Intercept: Horizontal Asymptote:
Ex. 3: Sketch each of the given functions
a) 1( ) 3xh x
b) ( ) 5 2xh x
74 AMHS Precalculus - Unit 5
The Natural Base e
Use you calculator to explore 1
lim(1 )n
n n.
Conclusion:
Ex. 4: Sketch each of the given functions
a) ( ) xf x e
b) ( ) xh x e
c) ( ) 2 xf x e
For c) State the Domain and Range
75 AMHS Precalculus - Unit 5
Ex. 5: Solve
a) 3 12 8x x
b) 2( 1)7 343x
c) 2 3 24 2x x d) 64 10(8 ) 16 0x x
Compound Interest
The amount of money ( )A t at some time t (in years) in an investment with an initial value, or principle
of P with an annual interest rate of r (APR – given as a decimal), compounded n times a year is:
( ) 1
ntr
A t Pn
Ex. 6: Determine the value of a CD in the amount of $1000.00 that matures in 6 years and pays 5% per year compounded
a) Annually
b) Monthly
c) Daily
76 AMHS Precalculus - Unit 5
Continuously Compounded Interest.
If the interest is compounded continuously ( n ), then the amount of money after t years is:
( ) rtA t Pe
Ex. 7: Determine the amount in the CD from example 6 if the interest is compounded continuously. Ex. 8: Which interest rate and compounding period gives the best return?
a) 8% compounded annually
b) 7.5% compounded semiannually
c) 7% compounded continuously Ex. 9: What initial investment at 8.5 % compounded continuously for 7 years will accumulate to $50,000?
77 AMHS Precalculus - Unit 5
Logarithmic Functions
Set up
Sketch ( ) 2xf x . Give the domain and range. Then find 1( )f x .
1( )f x =
Domain:
Range:
Intercept:
V.A.:
Definition
For each positive number 0a and each x in (0, ) , logay x if and only if yx a . yx a is the corresponding “exponential form” of the given “logarithmic form” logay x .
Ex. 1: Evaluate each expression.
a) 10log 1000
b) 10log 0.1
c) 2log 32
d) 2log 4
e) 8log 8
f) 3log 1
g) 2
5log 5 h) 3log 83
78 AMHS Precalculus - Unit 5
Properties of the Logarithm function with base a .
a) log 1 0a b) log 1a a
c) log x
a a x
d) loga xa x
Ex. 2: On the same coordinate plane, sketch the following functions.
( ) 3xf x and 3( ) logg x x
1( ) ( )
2
xf x and 1/2( ) logg x x
In general ….
( ) logag x x , 1a
Domain:
Range:
Intercept:
V.A.:
( ) logag x x , 0 1a
Domain:
Range:
Intercept:
V.A.:
79 AMHS Precalculus - Unit 5
Ex. 3: Sketch the following functions.
3( ) log ( 2)g x x
1/2( ) log 1g x x
The Natural Logarithm Function
The function defined by ( ) log lnef x x x and lny x iff yx e .
Ex. 4: On the same coordinate plane, sketch the following functions.
( ) xf x e and ( ) lng x x
80 AMHS Precalculus - Unit 5
Properties of the Logarithm function with base e .
a) b)
c) d)
Arithmetic Properties of Logarithms
For each positive number 1a , each pair of positive real numbers U and V , and each real number n
we have:
Base a Logarithm Natural Logarithm
a)
a) b)
b)
c)
c)
Ex. 5: Evaluate each expression
a) 4lne b) ln45e
c) 1
lne
d) (1/2)ln16e
e) 3ln8e
f) 2 2 2log 6 log 15 log 20
Change-of-Base Formula
For 0, 0, 0a a x ... log ln
loglog ln
a
x xx
a a
Ex. 6: Use your calculator to evaluate 6log 13 .
81 AMHS Precalculus - Unit 5
Ex.7: Use the properties of logarithms to simplify each expression so that the ln y does not contain
products, quotients or powers.
a) (2 1)(3 2)
4 3
x xy
x
b) 6 3 264 1 2y x x x
Solving Exponential and Logarithmic Equations
Ex. 8: Solve each of the given equations
a) 83xe b) 24 7xe
c) 2 13x
d) 2ln(3 ) 6x
e) ln( 1) ln( 3) 1x x
f) ln( 2) ln(2 3) 2lnx x x
82 AMHS Precalculus - Unit 5
g) 2log ( 3) 4x h) 1 5 32x xe
i) 2 6x x xe x e x e
j) ln 0x x x
Ex. 9: Given the function 3 1( ) 5xf x e , Find 1( )f x and state the domain and range of the inverse
function.
83 AMHS Precalculus - Unit 5
Exponential Growth and Decay
In one model of a growing (or decaying) population, it is assumed that the rate of growth (or decay) of
the population is proportional to the number present at time t (rate of growth = ( )kP t ). Using calculus,
it can be shown that this assumption gives rise to:
0( ) ktP t Pe where k is the rate of growth ( 0k ) or decay ( 0k ).
Ex.1: The number of a certain species of fish is given by 0.012( ) 12 tn t e where t is measured in years and
( )n t is measured in millions.
a) What is the relative growth rate of the population?
b) What will the fish population be after 15 years?
Ex.2: A bacteria culture starts with 500 bacteria and 5 hours later has 4000 bacteria. The population
grows exponentially.
a) Find a function for the number of bacteria after t hours.
b) Find the number of bacteria that will be present after 6 hours.
c) When will the population reach 15000?
84 AMHS Precalculus - Unit 5
Ex. 3: A culture of cells is observed to triple in size in 2 days. How large will the culture be in 5 days if the
population grows exponentially?
Ex. 4: Carbon-14, one of the three isotopes of carbon, has a half-life of 5730 years. If 10 grams were
present originally, how much will be left after 2000 years? When will there be 2 grams left?
Ex. 5: On September 19th, 1991, the remains of a prehistoric man were found encased in ice near the
border of Italy and Switzerland. 52.4% of the original carbon 14 remained at the time of the discovery.
Estimate the age of the Ice Man.
Ex. 6: The radioactive isotope strontium 90 has a half-life of 29.1 years.
a) How much strontium 90 will remain after 20 years from an initial amount of 300 kilograms?
b) How long will it take for 80% of the original amount to decay?
