Precalculus ... Do Now: Use the exponential properties to simplify and rewrite the
following expressions: .... Exponential functions as logarithms, log graphs Date: ...
Precalculus Prequiz- Logs and Exponential Functions
Name:____________________________________ Date:_____________________________________
MULTIPLE CHOICE (INDICATE YOUR ANSWERS IN THE SPACE PROVIDED): The expression (-3x2y3)3 is equivalent to:
(1)
(1) (a) -9x6y9
(b) -27x5y6
Simplify: (3 a )2 (3 a
(2)
4
(c) -27x6y9
(d) -3x5y6
) (2)
(a) Error! Bookmark not defined. 3 a 3
a
3
(3)
2
a
4
(b) 3 3a
4
(c) 3 2a
6
(d)
4a
What is log3 x
a written in exponential form? (3)
(a) 3x = a
(b) a3 = x
(c) Error! Bookmark not defined.ax = 3
=x
(4)
The equation y = ax expressed in logarithmic form is: (4) (a) x
(5)
loga y
(b) a
log x y
(c) x
log y a
(d) y
loga 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
(d) 3a
(7)
The expression log
a3 is equivalent to: b (7)
(a) 3
log a log b
1 (log a log b) 3 (c) 3(log a log b) (d) 3 log a log b
(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 log A log A log A
p 2 log r 2p(log r ) log p log 2 log r 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)
37 x 53 ( x 7)3 125
3x 3x
7
53 21 15
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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)
ax
(3)
ab
x
x
(4)
a b
(5)
ax ay
(6)
a
(7)
a0
y
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
2
9x
4
3
More Examples (each relates to “Process 2”): 4. 5. x 1 x 4 81
4
6.
(2 x 1)5
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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 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 1 3. e
Convert each into logarithmic form 4. ln 5 x
x
5. ln b
1
6. ln y
e
c
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 8. ln x y
x
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Precalculus
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Precalculus
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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) y log 4 ( x 1) y log 5 ( 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?
HW p313 #77-85 odd, 97-100 all 30
Precalculus
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Precalculus
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34
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Unit 6: Exponential & Logarithmic Functions Definitions, Properties & Formulas Properties of Exponents Property
Definition
Product
xa xb xa xb
Quotient
xa
b
xa
b
, where x
Power Raised to a Power
(xa)b = xab
Product Raised to a Power
(xy)a = xa ya a
x y
Quotient Raised to a Power
xa , where y ya
x0 = 1, where x
Zero Power x
Negative Power
1 , where x xn
n
0
0
0 0
1 n
Rational Exponent
n x 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
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 A
Compound Interest (Periodic)
Exponential Growth/Decay (in terms of e) Continuously Compounded Interest
P1
r 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
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Logarithmic Functions
are inverses of exponential functions a logarithm is an exponent! when no base is indicated, the base is assumed to be 10
Common Logarithms
log x
log10 x
log x
x
logb n logb a
loga n
Change of Base Formula
10 y
y
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
ey
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
written exponentially: b1 = b
1
logb b x
x
written exponentially: bx = bx
blog b
x , where x > 0
10log 10
x
logb MN
logb M logb N
7
7
log3 9 x
log3 9 log3 x
log 1 yz
log 1 y log 1 z
5
M logb N
p
logb M
logb M
5
2 5 7 log8 x log4
logb M logb N
p logb M
logb N
if and only if
log4 2 log4 5 log8 7 log8 x
log2 6 x
x log2 6
4
4 log5 y
log5 y
M=N
5
log6 (3 x 4) (3 x 4 )
log6 (5 x 2) ( 5 x 2)
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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
written exponentially: b1 = b
1
logb b x
x
written exponentially: bx = bx
blog b
x , where x > 0
10log 10
x
logb MN
logb M logb N
7
7
log3 9 x
log3 9 log3 x
log 1 yz
log 1 y log 1 z
5
5
2 5 7 log8 x log4
M logb N
logb M logb N
p
logb M
p logb M
logb M
logb N
if and only if
M=N
5
log4 2 log4 5
log8 7 log8 x
log2 6 x
x log2 6
log5 y 4
4 log5 y
log6 (3 x 4) (3 x 4 )
log6 (5 x 2) ( 5 x 2)
Common Errors: logb M logb N
logb M logb N
logb M logb N
logb (M N) p
(logb M)
logb M logb N p logb M
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
(logb M)p cannot be simplified
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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)
(2)
logb x 5 y
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)
(6)
log3 42
(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
(12) log7
(14) 10 x
1 49
log5 4 log5 x
75e 0.5 x
(11) log 6 x
x
(13) log x 4
(18) log9 (5 x )
1
(17) log4 x
1 2
1.002 4 x
(22) log( x 10) log( x 5)
(21) e 25 x 2
log 2 log( x 3)
3
(19) log 20 log x
3 log9 2
log5 36
2
(15) log x log 5
27.5
(16) log x log 2
(20) 2
(9)
1
1.25
(23) log 6 216
1 log 6 36 2
log 6 x
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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: Express the number of guppies as a function of the time t. Use your answer from part (a) to find how many guppies are present after 1 week? ce. Use your answer from part (a) to find, to the nearest day, when will there be 2,000 guppies? ac. bd.
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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.
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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.
(8) Jenny wants to buy a new flat screen television that will retain its value. She has a choice of buying the Luxury Brand television for $3,000 or the Expensive Brand television for $3,500. The Luxury Brand is assumed to have a depreciated value of $1,200 after three years, which represents a linear depreciation. The Expensive Brand depreciates in value exponentially at an average rate of 37.5% each year. (a) Write a linear function L(t) that relates the value of the Luxury Brand television in dollars to time t in years.
(b) Write an exponential function E(t) that relates the value of the Expensive Brand television in dollars to time t in years.
(c) If Jenny wants to be able to sell this flat screen television after 5 years, which model will have a higher value? Explain your answer.
(d) Sketch the graphs and indicate the WINDOW used.
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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!
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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-
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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? 45
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
Given:
Prove:
2
Isosceles triangle CAT, CT AT , ST bisects < CTA, SC and SA are drawn
