Precalculus
Prequiz- Logs and Exponential Functions
Name:____________________________________
Date:_____________________________________
MULTIPLE CHOICE (INDICATE YOUR ANSWERS IN THE SPACE PROVIDED):
(1)
The expression (-3x2y3)3 is equivalent to:
(1)
(a) -9x6y9
(2)
(b) -27x5y6
(c) -27x6y9
(d) -3x5y6
Simplify: (3 a )2 (3 a  4 )
(2)
(a) 3 a
(3)
2
a4
(b) 3 3 a  4
(c) 3 2a  6
(d) 3 a
3
 4a
What is log 3 x  a written in exponential form?
(3)
(a) 3x = a
(4)
(b) a3 = x
(c) ax = 3
(d) 3a = x
The equation y = ax expressed in logarithmic form is:
(4)
(a) x  log a y
(5)
(b) a  log x y
(c) x  log y a
(d) y  log a x
The expression log 12 is equivalent to:
(5)
(a)
(b)
(c)
(d)
(6)
log 3 + 2 log 2
log 6 + log 6
log 3  log 4
log 3 – 2 log 2
The expression log 4x is equivalent to:
(6)
(a)
(b)
(c)
(d)
4 log x
log 4 + log x
(log 4)(log x)
log x4
(7)
The expression log
a3
is equivalent to:
b
(7)
 log a 

(a) 3
 log b 
1
(b) (log a  log b)
3
(c) 3(log a  log b)
(d) 3 log a  log b
(8)
The expression log a 
1
log b is equivalent to:
2
(8)
(a) log( a  b )
(b) log a b
1

(c) log a  log b 
2

(d) log ab
(9)
If A = pr2, which equation is true?
(9)
(a)
(b)
(c)
(d)
(10)
log A  p  2 log r
log A  2p(log r )
log A  log p  log 2  log r
log A  log p  2 log r
Which of the following equations is equivalent to x log 3  7 log 3  3 log 5 ?
(10)
(a)
(b)
(c)
(d)
37x  53
( x  7)3  125
3 x  7  53
3x  21  15
2
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
3
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.
4
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.
5
__________________________________________________________________________________________
(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?
6
(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.
7
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
8
(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
9
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
10
__________________________________________________________________________________________
(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
11
(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.
12
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.
13
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)
14
(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.
15
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
16
__________________________________________________________________________________________
(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.
17
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?
18
(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
19
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
20
__________________________________________________________________________________________
(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.
21
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
Convert each into logarithmic form
1
4. log 25 5 
2
5. log a b  c
22
3.
1
6. log 3    2
9
1
 51
5
__________________________________________________________________________________________
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
7. ln y  x
x
8. ln x  y
23
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
24
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  b 3 )
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?
25
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
26
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)
27
Precalculus
Activity- Graphing Log Equations
**Will be collected and graded (separate paper)
Name:____________________________________
Date:_____________________________________
Objective:
To learn how to graph log equations that are not of base 10 or e.
DO NOW:
Find log 3 7 to the nearest ten-thousandths place.
__________________________________________________________________________________________
I.
1.
2.
3.
4.
Graph each of the following on the same set of coordinate axes and answer the following questions.
y  log 2 ( x  1)
y  log 3 ( x  1)
a.
What are some notable similarities and differences among the graphs?
b.
What appears to happen as the base gets larger and larger?
II.
1.
2.
3.
4.
Graph each of the following on the same set of coordinate axes and answer the following questions.
y  log 2 ( x  1)
y  log 2 ( x  2)
y  log 2 ( x  3)
y  log 2 ( x  4)
a.
What are some notable similarities and differences among the graphs?
b.
What appears to happen as the constant in the binomial changes?
y  log 4 ( x  1)
y  log 5 ( x  1)
HW p313 #77-85 odd, 97-100 all
28
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.
29
__________________________________________________________________________________________
(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?
30
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.
31
__________________________________________________________________________________________
(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?
32
__________________________________________________________________________________________
(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.
33
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
34
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
log 8  log 8 7  log 8 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)
35
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
log 8  log 8 7  log 8 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
36
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
log b 7
u
(3)
log b
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
37
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?
38
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.
39
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.
40
Precalculus
Activity- Review of Expoenentials and Logs
Name:__________________________
Date:___________________________
The accompanying diagrams contain exponential and logarithmic expressions and equations. When cut out, the
18 equilateral triangles fit together to form a large rhombus. For the triangles to create this shape, two
expressions that are equivalent must be touching each other, sharing the same edge. All triangles must be used
to complete the rhombus. There are expressions that have either the same or similar answers, so check your
work and each pairing carefully; otherwise you may find triangles that do not fit properly.
SHOW ALL WORK ON A SEPARATE SHEET OF PAPER!
41
Precalculus
Test x 2 - Winter Project
Name:____________________________________
Date:_____________________________________
Objective:
To use exponential & log functions to design a plan to save $1 million as quickly as possible.
Research
You will need:
~Job title, description, and salary
~Savings (assume interest rate is constant)/Investment Information
~Living expenses (including utilities, phone, groceries, entertainment, etc)
~Place to live (and amount of rent and renter’s insurance or mortgage and taxes
~Car/transportation expenses
~Miscellaneous expenses
~Prior debt (student loans, etc)
Math
You will need to include:
~Written explanation of your scenario (typed, double spaced, 12 point TNR font)
~exponential and logarithmic equations and their solutions or TVM Solver Data
~Graphs that model the rate of profit/income growth
~Written conclusion discussing the viability of your scenario
Due
Friday January 9, 2009
You will have (2) class sessions before the due date during which you may conduct research, ask questions of
me, conduct mathematical computations, and/or work on the verbal portion of the project.
We will also have (2) class sessions in a computer lab where we will:
1.
Learn how to create MS Word documents consisting of mathematical equations
2.
Be able to conduct research for our projects.
IDEAS?
Student LoansSavings AccountsTransportationOwn/Rent HouseInsuranceJobsMiscellaneous-
42
Precalculus
Model- Winter Project Calculations (Bland)
Name:____________________________________
Date:_____________________________________
Job title, description, and salary:
Savings:
Math Teacher, Teach Math, $4750 per month (used Median career value)
4% Savings account, deposit income – expense each month
Expenses that don’t go away
Electric, Gas, Oil:
Phone:
Groceries:
Entertainment:
Rent (including insurance):
Car expenses (maintenance):
Car insurance:
$400 per month
$75 per month
$500 per month
$350 per month
$1500 per month
$50 per month
$100 per month
Expense that expires after 5 years
Car Payment:
$333 per month
Expense that expires after 20 years
Student loans:
$373 per month
Prior Savings
$50,000
I had two expenses that did not carry on forever. Therefore I decided to break my project up into phases.
Phase I
0  t  5 Years
Income Expenditures
$4750 $400
$75
$500
$350
$1500
$50
$100
$333
$373
Surplus of $1069/month
Phase I:
Phase II:
Phase II
5  t  20 Years
Income Expenditures
$4750 $400
$75
$500
$350
$1500
$50
$100
$373
Phase III
t  20 Years
Income Expenditures
$4750 $400
$75
$500
$350
$1500
$50
$100
Surplus of $1402/month
Surplus of $1775/month
After 5 years, I now have a total of $131,923.4374 saved
After 20 years, I now have a total of $585,159.3123 saved
How long will it take me to arrive at a savings of $1,000,000?
I solved for N in TVM SOLVER and arrived at approximately 94.8568814 months beyond the 20th year. This
gives me a total of approximately 27 years, 10 months, 25 days, 16 hours, 57 minutes and 16 seconds to arrive
at $1,000,000 based on this information.
**Note- Two very important things to be aware of: a. I never got a raise! Do you think you might? How
much? When? and b. The costs in my scenario never increased! What about inflation? Higher taxes, etc?
43
44
Precalculus
Lesson- Math on MSWORD
Name:____________________________________
Date:_____________________________________
Objective:
To learn to use Microsoft Word to create math related documents.
Example:
1.
2.
y  2( x  2) 2  2
Given:
Prove:
Isosceles triangle CAT,
CT  AT ,
ST bisects < CTA,
SC and SA are drawn
<SCA  SAC
On Your Own:
( y  2) 2 ( x  2) 2

1
25
16
3
3  2
4 C3   

5  5
1
 x 5
x
Exit Ticket- MSWORD
**Print out and hand in at the end of class
1.
x 2  12 x  36 x 2  36

x6
6
2.
tan 590 =
c
15
45
Statistics
Lesson/HW- TVM Solver
Name:____________________________________
Date:_____________________________________
Objective:
To learn how to use the TVM Solver on the TI-83/84 to determine exponential growth and decay
as they apply to:
 Savings accounts
 Mortgages
 Student loan repayment
__________________________________________________________________________________________
Compound Interest Formula:
Mini Example:
_________________________________________________________________________________________
Fields in the TVM
The variables listed in the TVM solver are called 'fields'. Each variable represents a quantity associated with a
common finanial concept or formula.








N: This represents the number of compounding periods in the term of the investment, annuity or loan.
This will always be a positive value.
I%: This represents the 'nominal rate' for an investment, annuity or loan. This will always be a positive
value. Note: We write the percent form here, not the fraction or decimal form of a percent.
PV: This represents the 'present value' of an investment, loan or annuity. This number can be positive or
negative. If the number is positive, then it indicated money was collected as in a loan. If the number is
negative, then it represents money we paid out, as in an investment or loan where we are the lender.
PMT: This represents the payment made to build an annuity or pay off a loan. The value will always be
negative in these situations. If we have a 'payout' annuity, then the value will be positive. In either case,
the value represents the payment per compounding period.
FV: This represents the 'future value' of an investment, annuity or loan after N compounding periods
have passed. This value will be positive or negative depending on the signs of PV and PMT.
P/Y: This value represents the number of payments per year for annuities and loans.
C/Y: This represents the number of compounding periods per year. These must both be positive integers
greater than 1.
PMT: END BEGIN. This field allows one to set the TVM Solver for 'ordinary' annuities, (END), or
annuities 'due' (BEGIN).
46
Ex 1: Sue Simmons wants to re-finance her house. She currently owes $120,000 and closing costs will be
$4,500. She gets a 30-year mortgage at 6% nominal interest.
How large will her monthly payment be?
N=
I%=
PV=
PMT=
FV=
P/Y=
C/Y=
PMT: END BEGIN
Ex 2: Mike makes an initial deposit in a new savings account of $10,000. If this account accrues interest at a
rate of 3.9% compounded monthly and Mike deposits $500 per month, how many years (to the nearest
month) will it take him to have $1,000,000 in his account?
N=
I%=
PV=
PMT=
FV=
P/Y=
C/Y=
PMT: END BEGIN
Ex 3: Professor X had $50,000 in outstanding student loans at a 6.5% interest rate upon finishing grad school.
If he plans on paying the loan off in 10 years, what will his monthly payment be? How much on total
interest will he have paid at the end of the 10 years?
N=
I%=
PV=
PMT=
FV=
P/Y=
C/Y=
PMT: END BEGIN
On Your Own
1.
Duke plans on purchasing a 3BR house in Scarsdale for $700,000. He takes out a mortgage for
$750,000 to pay for realtor expenses, the first few months of utilities and taxes, and for some minor
cosmetic work on the house. The mortgage he qualifies for is a 30 year loan at 9% nominal interest.
What will his monthly payment be if he somehow got away with putting $0 as a down payment? What
will his monthly payment be if he put $100,000 as a down payment? How much in total interest will be
paid over the life of the 30 year mortgage in each case?
N=
I%=
PV=
PMT=
FV=
P/Y=
C/Y=
PMT: END BEGIN
N=
I%=
PV=
PMT=
FV=
P/Y=
C/Y=
PMT: END BEGIN
47
2.
Revisit example 3 from the lesson. Professor X decides to consolidate his loans over a 20 year period at
6% interest. How much more in interest will he have paid than on the 10 year plan?
N=
I%=
PV=
PMT=
FV=
P/Y=
C/Y=
PMT: END BEGIN
Extra Credit- 5 Test Points (all or nothing)
Suppose you currently live with your parents but would like to purchase a house of your own. You add
up all your current monthly expenses and subtract them from your monthly net salary and discover a
$1,500 surplus. You also have $40,000 in a savings account accruing interest at a 3.15% rate. You
deposit your surplus of $1,500 each month for a year before purchasing a house. You apply for a 30
year mortgage and get approved for $550,000 at a 8.3% interest rate. You are unsure if you can afford a
house that costs $550,000. Use the TVM Solver to determine how much of a mortgage you can afford
to take out.
Savings Account TVM
N=
I%=
PV=
PMT=
FV=
P/Y=
C/Y=
PMT: END BEGIN
Mortgage TVM
N=
I%=
PV=
PMT=
FV=
P/Y=
C/Y=
PMT: END BEGIN
48