Precalculus
Lesson- properties, equations with exponents and
power and exponential functions
Objectives:



Name:__________________________________
Date:___________________________________
use the properties of exponents
solve equations containing rational exponents
examine power and exponential functions
Do Now: Use the exponential properties to simplify and rewrite the following expressions:
(1)
ax  ay 
(2)
a 
(3)
ab
(4)
a
  
b
(5)
ax

ay
(6)
a x 
(7)
a0 
x y
x


x
__________________________________________________________________________________________
In Small Groups: Use each example in the “Do Now” to arrive at general rules as they apply to monomials with
exponents.
Using Exponential Function Properties to Solve for x:
Process 1
Examples (each relates to “Process 1”):
1.
2.
44 x1  42 x2
45 x1  16 2 x1
Process 2
3.
3x  9 x4
2
More Examples (each relates to “Process 2”):
x 4  81
4.
5.
x 1  4
6.
(2 x  1)5  32
Power function:
exponential function:
Small Group Activity
On your graphing calculator, simultaneously graph: y = 0.5x, y = 0.75x, y = 2x, y = 5x
(1)
What is the range of each exponential function?
(2)
What is the behavior of each graph?
(3)
Do the graphs have any asymptotes?
(4)
(a) What point is on the graph of each function?
(b) Why?
Characteristics of graphs of y = nx
n>1
0<n<1
domain
range
y-intercept
behavior
horizontal asymptote
vertical asymptote
Extension: Graph the exponential functions y = 2x, y = 2x + 3, and y = 2x – 2 on the same set of axes.
Compare and contrast the graphs using a table similar to the one above.
2
Precalculus
Lesson- Graphing exponential functions, exponential
growth and decay
Objectives:


Name:__________________________________
Date:___________________________________
graph exponential functions
use exponential functions to determine growth and decay
Using Exponential Functions for Real World Applications:
Exponential growth:
Exponential decay:
Exponential Growth or Decay: N = N0 (1 + r)t
(1) Write a formula that represents the average growth of the population of a city with a rate of 7.5% per year.
Let x represent the number of years, y represent the most recent total population of the city, and A is the
city’s population now. What is the expected population in 10 years if the city’s population now is 22,750
people? Graph the function for 0  x  20.
3
__________________________________________________________________________________________
(2) Suppose the value of a computer depreciates at a rate of 25% a year. Determine the value of a laptop
computer two years after it has been purchased for $3,750.
(3) Mexico has a population of about 100 million people, and it is estimated that the population will double in
21 years. If population growth continues at the same rate, what will be the population in:
(a) 15 years
(b) 30 years
(c) graph the population growth for 0  time  50
__________________________________________________________________________________________
(4) A researcher estimates that the initial population of honeybees in a colony is 500. They are increasing at a
rate of 14% per week. What is the expected population in 22 weeks?
4
(5) In 1990, Exponential City had a population of 700,000 people. The average yearly rate of growth is 5.9%.
Find the projected population for 2010.
(6) Find the projected population of each location in 2015:
(a)
In Honolulu, Hawaii, the population was 836,231 in 1990. The average yearly rate of growth is
0.7%.
(b)
The population in Kings County, New York has demonstrated an average decrease of 0.45% over
several years. The population in 1997 was 2,240,384.
5
Precalculus
Lesson- More exponential function graphs,
Population growth, half-life
Objectives:


Name:____________________________________
graph exponential functions
use exponential functions to determine population growth and half-life decay
(1) The population of Los Angeles County was 9,145,219 in 1997. If the average growth rate is 0.45%, predict
the population in 2010.
Graph the equation for 0  time  20.
(2) Radioactive gold 198 (198Au), used in imaging the structure of the liver, has a half-life of 2.67 days. If the
initial amount is 50 milligrams of the isotope, how many milligrams (rounded to the nearest tenth) will be
left over after:
(a) ½ day
(b) 1 week
6
(3) If a farmer uses 25 pounds of insecticide, assuming its half-life is 12 years, how many pounds (rounded to
the nearest tenth) will still be active after:
(a) 5 years
(b) 20 years
(4) In 2000, the chicken population on a farm was 10,000. The number of chickens increased at a rate of 9%
per year. Predict the population in 2005.
Graph the equation for 0  time  15.
(5) If Kenya has a population of about 30,000,000 people and a doubling time of 19 years and if the growth
continues at the same rate, find the population (rounded to the nearest million) in:
(a) 10 years
(b) 30 years
7
Precalculus
Lesson- Compound Interest
Name:__________________________________
Date:___________________________________
Objectives:

use exponential functions to determine compound interest
Do Now:
(1) A laser printer was purchased for $300 in 2001. If its value depreciates at a rate of 30% a year, determine
how much it will be worth in 2007.
(2) Rates can be compounded in different increments per year. Exponential growth occurs how often if the
rate is compounded:
annually:
bi-annually:
quarterly:
monthly:
weekly:
daily:
The general equation for exponential growth is modified for finding the balance in an account that earns
compound interest.
r

Compound Interest: A  P1  
n

nt
8
__________________________________________________________________________________________
(1) If Charlie invested $1,000 in an account paying 10% compounded monthly, how much will be in the
account at the end of 10 years?
(2) Mike would like to have $20,000 cash for a new car 5 years from now. How much should be placed in an
account now if the account pays 9.75% compounded weekly?
(3) Suppose $2,500 is invested at 7% compounded quarterly. How much money will be in the account in:
(c) ¾ year
(d) 15 years
__________________________________________________________________________________________
(4) Suppose $4,000 is invested at 11% compounded weekly. How much money will be in the account in:
(e) ½ year
(f) 10 years
9
(5) How much money must Cindy invest for a new yacht if she wants to have $50,000 in her account that earns
5% compounded quarterly after 15 years?
(6) Carol won $5,000 in a raffle. She would like to invest her winnings in a money market account that
provides an APR of 6% compounded quarterly. Does she have to invest all of it in order to have $9,000 in
the account at the end of 10 years? Show your work and explain your answer.
10
Precalculus
Lesson: Exponential Functions with base e
Name:__________________________________
Date:___________________________________
Objective:

use exponential functions with base e
Euler Savings Bank provides a savings account that earns compounded interest at a rate of 100%. You may
choose how often to compound the interest, but you can only invest $1 over the course of one year.
11
Exponential Growth or Decay (in terms of e): N = N0 ekt
(1)
According to Newton, a beaker of liquid cools exponentially when removed from a source of heat.
Assume that the initial temperature Ti is 90F and that k = 0.275.
(a) Write a function to model the rate at which the liquid cools.
(b) Find the temperature T of the liquid after 4 minutes (t)
(c) Graph the function and use the graph to verify your answer in part (b)
12
(2)
Suppose a certain type of bacteria reproduces according to the model
time in hours.
B = 100 e0.271 t , where t is the
(a) At what percentage rate does this type of bacteria reproduce?
(b) What was the initial number of bacteria?
(c) Find the number of bacteria (rounded to the nearest whole number) after:
(i) 5 hours
(ii) 1 day
(iii) 3 days
(3)
A city’s population can be modeled by the equation y = 33,430e0.0397 t , where t is the number of years
since 1950.
(a) Has the city experienced a growth or decline in population?
(b) What was the population in 1950?
(c) Find the projected population in 2010.
13
Precalculus
HW- Compound Interest
Name:__________________________________
Date:___________________________________
(1) If you invest $5,250 in an account paying 11.38% compounded continuously, how much money will be in
the account at the end of:
(a) 6 years 3 months
(b) 204 months
(2) If you invest $7,500 in an account paying 8.35% compounded continuously, how much money will be in
the account at the end of:
(a) 5.5 years
(b) 12 years
14
__________________________________________________________________________________________
(3) A promissory note will pay $30,000 at maturity 10 years from now. How much should you be willing to
pay for the note now if the note gains value at a rate of 9% compounded continuously?
(4) Suppose Niki deposits $1,500 in a savings account that earns 6.75% interest compounded continuously.
She plans to withdraw the money in 6 years to make a $2,500 down payment on a car. Will there be
enough funds in Niki’s account in 6 years to meet her goal? Explain your answer.
15
Precalculus
Lesson- Continuous Compound Interest
Name:__________________________________
Date:___________________________________
Objective:

use exponential functions to determine continuously compounded interest
Continuously Compounded Interest: A = Pert
(1) Tim and Kerry are saving for their daughter’s college education. If they deposit $12,000 in an account
bearing 6.4% interest compounded continuously, how much will be in the account when she goes to college
in 12 years?
(2) Paul invested a sum of money in a certificate of deposit that earns 8% interest compounded continuously.
If Paul made the investment on January 1, 1995, and the account was worth $12,000 on January 1, 1999,
what was the original amount in the account?
16
(3) Compare the balance after 30 years of a $15,000 investment earning 12% interest compounded
continuously to the same investment compounded quarterly.
(4) Given the original principal, the annual interest rate, the amount of time for each investment, and the type
of compounded interest, find the amount at the end of the investment:
(a)
P = $1,250;
r = 8.5%;
t = 3 years;
compounded semi-annually
(b)
P = $2,575;
r = 6.25%;
t = 5 years 3 months;
compounded continuously
17
Precalculus
HW- Compound Interest
Name:__________________________________
Date:___________________________________
(1) If you invest $5,250 in an account paying 11.38% compounded continuously, how much money will be in
the account at the end of:
(a) 6 years 3 months
(b) 204 months
(2) If you invest $7,500 in an account paying 8.35% compounded continuously, how much money will be in
the account at the end of:
(a) 5.5 years
(b) 12 years
18
__________________________________________________________________________________________
(3) A promissory note will pay $30,000 at maturity 10 years from now. How much should you be willing to
pay for the note now if the note gains value at a rate of 9% compounded continuously?
(4) Suppose Niki deposits $1,500 in a savings account that earns 6.75% interest compounded continuously.
She plans to withdraw the money in 6 years to make a $2,500 down payment on a car. Will there be
enough funds in Niki’s account in 6 years to meet her goal? Explain your answer.
19
Precalculus
Lesson- Properties of a logs, rewriting
Exponential functions as logarithms, log graphs
Name:____________________________________
Date:_____________________________________
Objective:
 To learn what a logarithm is
 To learn the properties of logs
 To learn to rewrite an exponential function as a logarithm
 Graphing logs

Do Now:
Solve for x: 3x  9 x1 and check.
_________________________________________________________________________________________
What is a logarithm?
Logarithms are inverses of exponential functions. Logarithms are functions because exponential functions are
one-to-one functions.
We cannot solve an equation like: y  2 x using the algebraic techniques we have learned so far. Therefore, we
must try an alternative technique.
Rule: x  b y is equivalent to y  log b x
The log to the base b is the exponent to which b must be raised to obtain x.
Properties of Logs
logb 1  0
logb b  1
logb b x  x
blog b x  x , where x > 0
logb MN  logb M  logb N
M
logb  logb M  logb N
N
logb Mp  p logb M
Example:
Convert each into logarithmic form
1. y  2 x
2. 3  9
3.
1
 51
5
Convert each into logarithmic form
1
4. log 25 5 
2
5. log a b  c
1
6. log 3    2
9
20
_________________________________________________________________________________________
What is a Natural Logarithm?
Rule: x  b y is equivalent to y  log b x
The log to the base b is the exponent to which b must be raised to obtain x.
Properties of Logs
ln 1  0
ln b  1
ln e x  x
e ln x  x , where x > 0
ln MN  ln M  ln N
M
ln
 ln M  ln N
N
ln M p  p ln M
Example:
Convert each into logarithmic form
1. y  e x
2. e  x
1
 e 1
3.
e
Convert each into logarithmic form
4. ln 5  x
5. ln b  c
6. ln y  2
Example:
Graph each of the following on the same set of axes using the graphing calculator. y
1. y  2 x
2. x  2 y
3. log 2 y  x
4. log 2 x  y
5. y  e x
6. x  e y
x
7. ln y  x
8. ln x  y
21
Precalculus
Name:__________________________________
Lesson/HW- Simplify log expressions, common logs, evaluate
Date:___________________________________



Objectives:
simplify expressions using the properties of logarithmic functions
define common logarithms
evaluate expressions involving logarithms
Problem Set: write the following expressions in simpler logarithmic forms:
(1)
logb u2 v 7
(2)
logb
1
a2
(4)
log b
u
vw
(6)
logb
2
(3)
log b
m3
n
1
2
3
n
p q3
(5)
logb x
(7)
Use logarithmic properties to find the value of x (without using a calculator):
1
2
logb x  logb 9  logb 8  logb 6
2
3
2
22
Write each expression in terms of a single logarithm with a coefficient of one:
(8)
5 logb x  4 logb y
(10) 3 logb x  2 logb y 
(12)
(9)
1
logb z
4
ie : 2 log b u  log b v  log b
u2
v
2 logb x  logb y
(11)  8 logb c
3
logb w  2 logb u
2
(13)
1
logb (a 2  b3 )
3
Common Logarithm:
log 10  log x
Change of Base Formula:
log b a 
log a ln a log p a


log b ln b log p b
Given loga n, evaluate each logarithm to four decimal places:
(14) log8 172
(15) log 6 1.258
(16) log13 0.0065
Extension: Given y = logb n, what can you determine about the log value (y) based on b and n?
23
Precalculus
Lesson/HW- Properties of Logarithmic Functions,
Simplifying logarithmic expressions
Objective:


Name:____________________________________
Date:_____________________________________
examine properties of logarithmic functions
simplify expressions using the properties of logarithmic functions
Use the properties of logarithmic functions to solve for x:
(1)
log5 x  2
(2)
log4 64  x
(3)
logx 8  3
(4)
log8 x 
2
3
Use the properties of logarithmic functions to simplify each expression:
(5)
log8 8
(6)
log0.5 1
(7)
log10 1,000
(8)
log2 64
(9)
log7 343
(10)
log10 0.001
(11)
loge e
(12)
log5 3 5
24
Write the following expressions in simpler logarithmic forms:
(14)
logb
v7
u8
mn
pq
(16)
logb
1
a4
logb 5 x
(18)
logb 3 x 2  y 2
(13)
logb x 6 y 9
(15)
logb
(17)
25
Precalculus
Lesson- Natural Log Word Problems
Name:__________________________________
Date:___________________________________
Objectives:

solve real-world applications with natural logarithmic functions
Do Now:
Laura won $2,500 on a game show. She would like to invest her winnings in an account that earns an interest
rate of 12% compounded continuously. Does she have to invest all of it in order to have $4,000 in the account
at the end of 4 years to put a down payment on a new sailboat? Show your work and explain your answer.
(1)
Ana is trying to save for a new house. How many years, to the nearest year, will it take Ana to triple the
money in her account if it is invested at 7% compounded annually?
(2)
At what annual percentage rate (to the nearest hundredth of a percent) compounded continuously will
$6,000 have to be invested to amount to $11,000 in 8 years.
26
__________________________________________________________________________________________
(3) In 1990, Exponential City had a population of 142,000 people. In what year will the city have a
population of about 200,000 people if it was growing at an exponential rate of k = 0.014?
(4)
If $5,000 is invested at an annual interest rate of 5% compounded quarterly, how long will it take the
investment to double?
(5)
What was the annual interest rate (to the nearest hundredth of a percent) of an account that took 12 years
to double if the interest was compounded continuously and no deposits or withdrawals were made during
the 12-year period?
27
Precalculus
Lesson- More natural log word problems
Name:__________________________________
Date:___________________________________
Objective:

solve real-world applications with natural logarithmic functions
(1) If a car originally costs $18,000 and the average rate of depreciation is 30%, find the value of the car to the
nearest dollar after 6 years.
(2) How many years, to the nearest year, will it take for the balance of an account to double if it is gaining 6%
interest compounded semiannually?
(3) When Rachel was born, her mother invested $5,000 in an account that compounded 4% interest monthly.
Determine the value of this investment when Rachel is 25 years old.
28
__________________________________________________________________________________________
(4) The decay of carbon-14 can be described by the formula A  A 0 e 0.000124 t . Using this formula, how many
years, to the nearest year, will it take for carbon-14 to diminish to 1% of the original amount?
(5) In 2002, a farmer had 400 pigs on his farm. He estimated that this population of pigs will double in 15
years. If population growth continues at the same rate, predict the number of pigs in:
a. 2010
b. 2030
(6) If the world population is about 6 billion people now and if the population grows continuously at an annual
rate of 1.7%, what will the population be (to the nearest billion) in 10 years from now?
29
__________________________________________________________________________________________
(7) If $100 is invested in an account that has an interest of 7% compounded quarterly, how long will it take for
the balance to reach a value of $1,000?
(8) What interest rate (to the nearest hundredth of a percent) compounded monthly is required for an $8,500
investment to triple in 5 years?
(9) An optical instrument is required to observe stars beyond the sixth magnitude, the limit of ordinary vision.
However, even optical instruments have their limitations. The limiting magnitude L of any optical
telescope with lens diameter D, in inches, is given by the equation L  8.8  5.1 log D . Use this equation
to find the following to the nearest tenth:
a.
the limiting magnitude for a homemade 6-inch reflecting telescope.
b. the diameter of a lens that would have a limiting magnitude of 20.6.
30
Unit 6: Exponential & Logarithmic Functions
Definitions, Properties & Formulas
Properties of Exponents
Property
Definition
Product
x a xb  xa  b
Power Raised to a Power
xa
 x a  b , where x  0
b
x
(xa)b = xab
Product Raised to a Power
(xy)a = xa ya
Quotient Raised to a Power
x
xa
   a , where y  0
y
y
Zero Power
x0 = 1, where x  0
Quotient
a
Negative Power
x n 
1
, where x  0
xn
1
n
Rational Exponent
x n x
for any real number x  0 and any integer n > 1
and when x < 0 and n is odd
N = N0 (1 + r)t
Exponential
Growth/Decay
Compound
Interest (Periodic)
Exponential
Growth/Decay
(in terms of e)
Continuously
Compounded
Interest
where: N is the final amount, N0 is the initial amount, t is the number of time
periods, and r is the average rate of growth(positive) or decay(negative) per
time period
r

A  P1  
n

nt
where: A is the final amount, P is the principal investment, r is the annual
interest rate, n is the number of times interest is compounded each year, and t is
the number of years
N = N0 ekt
where: N is the final amount, N0 is the initial amount, t is the number of time
periods, and k (a constant) is the exponential rate of growth(positive) or
decay(negative) per time period
A = Pert
where: A is the final amount, P is the principal investment, r is the annual
interest rate, and t is the number of years
31
Logarithmic
Functions
are inverses of exponential functions
 a logarithm is an exponent!
when no base is indicated, the base is assumed to be 10
log x  log10 x
Common
Logarithms
 log x  y  10 y  x
Change of Base
Formula
loga n 
logb n
logb a
where a, b, and n are positive numbers, and a 1, b 1
instead of log, ln is used; these logarithms have a base of e
Natural
Logarithms
ln x  loge x
 ln x = y  e y  x
all properties of logarithms also hold for natural logarithms
Properties of Logarithmic Functions
If b, M, and N are positive real numbers, b  1, and p and x are real numbers, then:
Definition
Examples
logb 1  0
written exponentially: b0 = 1
logb b  1
written exponentially: b1 = b
logb b x  x
written exponentially: bx = bx
blog b
x
 x , where x > 0
10log 10 7  7
logb MN  logb M  logb N
log3 9x  log3 9  log3 x
log 1 yz  log 1 y  log 1 z
5
logb
2
 log 4 2  log 4 5
5
7
log8  log8 7  log8 x
x
log2 6 x  x log2 6
logb M  p logb M
p
if and only if
5
log 4
M
 logb M  logb N
N
logb M  logb N
5
log 5 y 4  4 log 5 y
M=N
log6 (3x  4)  log6 (5x  2)
 ( 3 x  4 )  ( 5 x  2)
32
Properties of Logarithmic Functions
If b, M, and N are positive real numbers, b  1, and p and x are real numbers, then:
Definition
Examples
logb 1  0
written exponentially: b0 = 1
logb b  1
written exponentially: b1 = b
logb b x  x
written exponentially: bx = bx
blog b x  x , where x > 0
10log 10 7  7
logb MN  logb M  logb N
log3 9x  log3 9  log3 x
log 1 yz  log 1 y  log 1 z
5
logb
2
 log 4 2  log 4 5
5
7
log8  log8 7  log8 x
x
log2 6 x  x log2 6
logb M  p logb M
p
if and only if
5
log 4
M
 logb M  logb N
N
logb M  logb N
5
log 5 y 4  4 log 5 y
M=N
log6 (3x  4)  log6 (5x  2)
 ( 3 x  4 )  ( 5 x  2)
Common Errors:
logb M
 logb M  logb N
logb N
logb (M  N)  logb M  logb N
(log b M)  p logb M
p
logb M  logb N  logb
M
N
logb M
cannot be simplified
logb N
logb M  logb N  logb MN
logb (M  N) cannot be simplified
p logb M  logb Mp
(log b M)p cannot be simplified
33
Precalculus
Review- Exponential and Logarithmic Functions part 1
Name:____________________________________
Date:_____________________________________
ANSWER THE FOLLOWING QUESTIONS ON A SEPARATE SHEET OF PAPER AND SHOW ALL WORK!
Write each expression in terms of simpler logarithmic forms:
4
(1)
log b x 5 y
(2)
s5
logb 7
u
(3)
logb
1
c8
(4)
logb
m 5n 3
p
Given loga n, evaluate each logarithm to four decimal places:
(5)
log3 42
(6)
(7)
log 1 5
log6 0.00098
2
Solve each equation and round answers to four decimal places where necessary:
(8)
log2 x  3
(10) 1000  75e0.5 x
(12) log 7
1
x
49
(9)
log5 4  log5 x  log5 36
(11) log 6 x  2
(13) log x 4 
1
2
(14) 10 x  27.5
(15) log x  log 5  log 2  log( x  3)
(16) log x  log 2  1
(17) log4 x  3
(18) log9 (5  x )  3 log9 2
(19) log 20  log x  1
(20) 2  1.002 4 x
(21) e 25 x  1.25
(22) log( x  10)  log( x  5)  2
(23) log 6 216 
1
log 6 36  log 6 x
2
34
Precalculus
Review- Exponential and Logarithmic Functions part 2
Name:____________________________________
Date:_____________________________________
SHOW ALL WORK:
(1) Anthony is an actuary working for a corporate pension fund. He needs to have $14.6 million grow to $22
million in 6 years. What interest rate (to the nearest hundredth of a percent) compounded annually does he
need for this investment?
(2)
The number of guppies living in Logarithm Lake doubles every day. If there are four guppies initially:
c.
Express the number of guppies as a function of the time t.
d.
Use your answer from part (a) to find how many guppies are present after 1 week?
e.
Use your answer from part (a) to find, to the nearest day, when will there be 2,000 guppies?
35
SHOW ALL WORK:
(3)
The relationship between intensity, i, of light (in lumens) at a depth of x feet in Lake Erie is given by
i
log
 0.00235 x . What is the intensity, to the nearest tenth, at a depth of 40 feet?
12
(4)
Tiki went to a rock concert where the decibel level was 88. The decibel is defined by the formula
i
D  10 log , where D is the decibel level of sound, i is the intensity of the sound, and i0 = 10 -12 watt per
i0
square meter is a standardized sound level. Use this information and formula to find the intensity of the
sound at the concert.
36
SHOW ALL WORK:
(5)
How many years, to the nearest year, will it take the world population to double if it grows continuously at
an annual rate of 2%.
(6)
Bank A pays 8.5% interest compounded annually and Bank B pays 8% interest compounded quarterly. If
you invest $500 over a period of 5 years, what is the difference in the amounts of interest paid by the two
banks?
(7)
Determine how much time, to the nearest year, is required for an investment to double in value if interest
is earned at the rate of 5.75% compounded quarterly.
37