chapter 6: logarithmic functions and systems of equations · the logarithm, correct to three ......

Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations

Chapter 6: Logarithmic Functions and Systems of Equations

QUIZ AND TEST INFORMATION: The material in this chapter is on

Quiz 6 and the final exam. You should complete all three attempts of Quiz

6 before taking the final exam.

TEXT INFORMATION: The material in this chapter corresponds to the

following sections of your text book: 4.4, 4.5, 4.6, 4.7, 4.8, 5.1, and 5.2.

Please read these sections and complete the assigned homework from the

text that is given on the last page of the course syllabus.

LAB INFORMATION: Material from these sections is used in the

following labs: Logarithmic Functions.

Logarithmic Functions Systems of Equations Comprehensive Final

Assignments 4.4-4.8 and 5.1-5.2

Lab Logarithmic Functions

Quiz 6

Test Comprehensive Final

176


Section 1: Logarithmic Functions

• Definition: We now define a function that is the inverse function to

an exponential function. If a (the b__________) is greater than

_______ and not equal to _________, then y =_____________ if and

only if ________________________________.

• Example 1: What is log28?

• Definition: The natural log is the inverse function of

______________. Thus, the natural log is log base _______. We use

the following special notation for the natural log ________________.

The common log is the inverse function of __________________.

Thus, the common log is log base _________. We use the following

notation for the common log _______________________.

• Example 2: Convert the following statements to logarithmic form.

1. y = 2x

2. 8 = 23

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• Example 3: Convert the following to exponential form. log39

• Example 4: Convert the following to exponential expressions and

evaluate if possible.

1. log21

2. log 10

3. log 3

4. log 0

5. loga1

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6. log a a

7. log a ax

• Example 5: Sketch the graph of f(x) = 2x, g(x) = log2x and y = x on

the same set of axes. Label the graphs.

• Properties of Graphs of Logarithmic Functions: For f(x)=logax where

a > 1, the following are properties of the graph of f(x):

1. The function is i_____________________.

2. The graph includes the points (1,_____) and (_____,1).

3. As x approaches infinity, f(x) approaches ________________.

4. As x approaches zero, f(x) approaches ________________.

5. Domain = _____________________________

6. Range = _______________________________

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• Example 6: Sketch the graphs of f(x) = (1/2)x and g(x) = log(1/2)x on

the same set of axes. Label the graphs.

• Properties of Graphs of Logarithmic Functions: For f(x) = logax

where 0 < a < 1, the following are properties of the graph of f(x):

1. The function is d_____________________.

2. The graph includes the points (1,_____) and (_____,1).

3. As x approaches infinity, f(x) approaches ________________.

4. As x approaches zero, f(x) approaches ________________.

5. Domain = _____________________________

6. Range = _______________________________

• Example 7: Convert ex = 21 to logarithmic form.

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• Example 8: Graph y = lnx, y = -lnx, and y = ln(-x) on the same set of

axes and label the graphs. Give the domain, range, and vertical

asymptote of each.

y = lnx Domain: Range: Vertical asymptote: y = -lnx Domain: Range: Vertical asymptote: y = ln(-x) Domain: Range: Vertical asymptote:

• Example 9: Find the domain of the function y = log2(1 - x).

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• Properties of logarithm:

1. loga1 = ________

2. log a a = ________

3. log a a r= _________

4. log a MN = _________________________________________

5. log a (M/N) = _______________________________________

6. log a M r = _________________________________________

• Example 10: Expand the following using the log properties.

1. log3(5x)

2. log3(3x2)

3. 12log +xa

4. ⎟⎟⎠

⎞⎜⎜⎝

⎛−1

3

log xx

a

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• Example 11: Solve y = log35.

• Change of Base Formula: If y = logax, then

a

xy

b

b

loglog

=

• Example 12: Use a calculator and the base conversion formula to find

the logarithm, correct to three decimal places. log9 71.30

• Example 13: Solve each of the following equations.

1. log3(3x - 2) = 2

183


2. -2 log4x = log49

3. 2log3(x + 4) - log39 = 2

4. 5 1-2x = 1/5

5. 242 xx =

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6. 3x = 14

7. log(4x) = log5 + log(x - 1)

8. log3(6x + 4) = log3(6x + 7)

• Other examples and notes:

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Section 2: Applications

• Definition: ____________________ interest is interest paid on the

amount borrowed or invested (does not include any interest that

accrues). ___________________ interest is applied yearly and is a

form of linear growth.

• Definition: ______________________ interest is interest paid on the

amount borrowed and previously owed interest.

_______________________ interest can be applied annually, semi-

annually, quarterly, monthly, daily, etc. and is a form of exponential

growth.

• Simple Interest Formula: I = Prt

1. P = ______________________

2. r = _______________________

3. t = _______________________

4. I = _______________________

• Compound Interest Formula: ⎟⎠⎞

⎜⎝⎛ +=

nr

nt

PA 1

1. P = ______________________

2. r = _______________________

3. t = _______________________

4. A = _______________________

5. n = _______________________

186


• Example 1: Let P = $100, t = 1, and r = 3%. What is the amount if

compounded quarterly? What is the amount if compounded monthly?

Compounded quarterly: Compounded monthly:

• Continuous Compounding Formula: A = Pert

1. A = _________________________

2. P = _________________________

3. r = __________________________

4. t = __________________________

• Example 2: What is the amount if compounded continuously when

P = $100, t = 1, and r = 3%?

187


• Definition: The Present Value of A dollars to be received at a future

date is the principal you would need to invest now so that it would

grow to A dollars in a specified time period.

• Example 3: Find the P needed to get $800 after 3.5 years at 7%

interest compounded monthly.

188


• Example 4: Eddie’s son will be going to college in 5 years. If Eddie

needs to save $30,000 over the next 5 years for his son to go to

college, how much should he invest now if he will get a 3.5% interest

rate compounded continuously?

189


• Example 5: How long will it take to double an investment if r = 7%

and it is compounded continuously?

190


• Law of Uninhibited Growth and Decay: If a quantity is changing

exponentially, then A = A0ekt where

1. A = ___________________

2. A0 = _____________________

3. t = _______________________

4. k = ______________________ (> 0 if growing, and < 0 if

decaying).

• Example 6: A colony of bacteria increases according to the law of

uninhibited growth. If the number of bacteria doubles in 3 hours, how

long will it take for the size of the colony to triple?

191


• Example 7: The half life of carbon 14 is 5600 years. If 10g are

present now, how much will be present in 100 years?


192


Section 3: Systems of Linear Equations

• Definition: A _______________________________ is a collection of

two or more equations in two or more variables. A

___________________ to a system of equations is a set of values for

the variables that make each equation in the system true. If you are

asked to _________________ a system of equation, then you should

find ________ solutions to the system.

• Example 1: Is (1, 2) a solution to the following system of equations?

2x – y = 0

2x – ½ y = 1

• Definition: When a system of equations has at least one solution, the

system is said to be ________________________________________.

Otherwise, the system is called ______________________________.

• Definition: An equation in n variables is said to be linear, if it is

equivalent to an equation of the form

________________________________________________________.

• Definition: If every equation in a system of equations is linear, then

we call the system of equations ______________________________.

193


• Note: The simplest system of equations to study is a system of two

linear equations in two variables. In this case, the graph of each

equation will be a _____________________. Then there are three

possible things that could happen.

1. The lines intersect. In this case, the system will have _____

solution given by the point of ________________________.

We call the equations _______________________________.

2. The lines are parallel. In this case, the system will have

__________________________________.

3. The lines are ________________________ (meaning they are

the same line). In this case, there are _____________________

solutions. We call the equations ________________________.

• Example 2: Solve the following system of equations by substitution.

x + 2y = 5

x + y = 3

• Example 3: Solve the following system of equations by substitution.

2x + y = 5

4x +2y = 8

194


• Example 4: Solve the following system of equations by elimination.

x + y = 5

2x – y = 4


x + 3y = 5

2x - y = 3

195



2x + y = 4

-6x - 3y = -12

• Note: Now let’s look at systems of equations with three equations in

three variables. As above, the systems could have exactly _______

solution, ________ solutions, or ____________________________

solutions.

196


• Example 7: Use the method of elimination to solve the following

system of equations.

x + y – z = 2

2x + y - 3z = 1

x - 2y + 4z = 5

197


• Example 8: The perimeter of a field is 300 feet. Find the dimensions

of the field if the length is half the width.


198


Section 4: Matrices

• Definition: A __________________ is a rectangular array of

numbers. Each entry in the matrix has two indices: a _________

index and a ________________ index.

• Example 1: Write some examples of matrices. Look at entries and

determine their row index and column index.

• Example 2: Write what a general matrix looks like.

199


• Note: If we write down a system of equations, but don’t write down

the ____________________, it is a matrix. Note that if a variable is

missing in an equation in a system, then its coefficient is ________.

Such a matrix is called an _____________________________ matrix.

• Example 3: Write a system of equations and its augmented matrix.

• Definition: Row operations are used to ________________ a system

of equations when we write the system as an augmented matrix. The

following are row operations.

1. interchanging any two rows

2. replacing a row by any nonzero multiple of that row

3. replacing a row by the sum of that row and a constant nonzero

multiple of another row

200


• Example 4: Write some matrices and perform row operations on

them.

201


• Definition: A matrix is in ___________________________________

when the following are true about the matrix.

1. The a11 entry is 1 and all entries below it are 0.

2. The first nonzero entry in each row after the first row is a 1,

zeros appear below it, and it appears to the right of the first

nonzero entry in any row above.

3. Any row containing all zeros, except possibly in the last entry,

appears at the bottom.

• Example 5: Write some examples of matrices in row echelon form

and some examples of matrices that are not in row echelon form.

In row echelon form:

Not in row echelon form:

202


• Note: To solve a system of equations using matrices, write the

augmented matrix that corresponds to the system. Use

____________________________ to put this matrix in row echelon

form. Analyze the system of equations that corresponds to this new

matrix to __________________ the original system.

• Example 6: Solve the following system of equations using matrices.

2x + 2y = 6

x + y + z = 1

3x + 4y – z = 13

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chapter 6: logarithmic functions and systems of equations · the logarithm, correct to three ......

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