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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
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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
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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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
• 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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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
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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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
• 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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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
• 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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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
• 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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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
• 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
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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
2. -2 log4x = log49
3. 2log3(x + 4) - log39 = 2
4. 5 1-2x = 1/5
5. 242 xx =
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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
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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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
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 = _______________________
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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
• 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%?
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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
• 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.
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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
• 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?
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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
• Example 5: How long will it take to double an investment if r = 7%
and it is compounded continuously?
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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
• 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?
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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
• Example 7: The half life of carbon 14 is 5600 years. If 10g are
present now, how much will be present in 100 years?
• Other examples and notes:
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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
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 ______________________________.
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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
• 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
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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
• Example 4: Solve the following system of equations by elimination.
x + y = 5
2x – y = 4
• Example 5: Solve the following system of equations by elimination.
x + 3y = 5
2x - y = 3
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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
• Example 6: Solve the following system of equations by elimination.
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.
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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
• 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
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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
• Example 8: The perimeter of a field is 300 feet. Find the dimensions
of the field if the length is half the width.
• Other examples and notes:
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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
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.
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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
• 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
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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
• Example 4: Write some matrices and perform row operations on
them.
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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
• 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:
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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
• 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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Copyright 2006 Melanie Butler Chapter 6: Logarithmic Functions and Systems of Equations
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• Other examples and notes: