MATH 30-1
Exponents & Logarithms
Module Six Assignment
Module / Unit 6 - Assignment Booklet
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Math 30-1: Module 6 Assignment
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Math 30-1: Module 6 Assignment
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Lesson 1: Exponential Functions
æ öx
1. Sketch the graph of y = çç 1 ÷
÷ and state the domain, range, horizontal asymptote, and
è3 ø
intercepts.
2. Complete these tasks for each of the following functions.
i.
Describe the transformations.
ii. Sketch the graph of the function.
iii. Identify the domain, range, equation of the horizontal asymptote, and intercepts.
x
a. y = 3 (2) + 1
Math 30-1: Module 6 Assignment
b. y = -
x
c.
x+ 2
(21)
y = (3)4 + 1
- 3
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Math 30-1: Module 6 Assignment
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3. The radioactive isotope Molybdenum-99 is produced at Chalk River Laboratories.
Molybdenum-99 has a half-life of 66 h.
a. What is the exponential function that would describe the relationship between the
percent, P, of Molybdenum-99 remaining after t hours?
b. What percent of Molybdenum-99 would be left after 24 h?
Math 30-1: Module 6 Assignment
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LESSON 1 SUMMARY
In this lesson you explored exponential functions and how they can be used to describe
growth or decay of quantities.
The characteristics of all exponential functions of the form y = cx, c > 0, c ≠ 1 are as
follows:
o
o
o
o
o
o
o
If c > 1, the function is increasing and is an exponential growth function.
If 0 < c < 1, the function is decreasing and is an exponential decay function.
The domain is {x|x ∈ R}.
The range is {y|y > 0, y ∈ R}.
There is no x-intercept.
The y-intercept is 1.
The horizontal asymptote is at y = 0.
When an exponential function is in the form y = a(c)b(x–h) + k, the parameters a, b, h
and k correspond to the following transformations:
Value > 0
o
vertical stretch of
the graph by a factor
of |a|
o
horizontal stretch of
the graph by a factor
a
b
Value < 0
o
vertical stretch of
the graph by a
factor of |a|, and a
reflection in the xaxis
o
horizontal stretch
of the graph by a
factor of
, and a
reflection in the yaxis
of
h
k
o
translated to the
right h units
o
translated up k units
o
translated to the
left |h| units
o
translated down |k|
units
Exponential functions can be used to model real-life applications of exponential growth
or decay. In the next lesson you will explore how to solve exponential equations
Math 30-1: Module 6 Assignment
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Lesson 2: Exponential Equations
x
2x 3
1. Write the expressions 36 and 216
x
so that both expressions have the same base.
3x4
2. Solve the exponential equation 32  8
by changing the bases. Check your answer by
graphing, and include a sketch of your graph.
Math 30-1: Module 6 Assignment
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x4
3. Solve the exponential equation 25
using substitution.
2x
 625 by changing the bases. Check your answer
4. Create your own exponential equation that contains different bases. Include a full solution
to your equation.
5. Trinh has decided to save money to buy a new computer. She invests $800 in an account
that earns 5.0% per year, compounded quarterly. Determine how long she needs to invest
her money until she has $1000. Explain the process you used.
Math 30-1: Module 6 Assignment
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LESSON 2 SUMMARY
In this lesson you explored how to solve exponential equations.
You learned that when there is a single power on each side of the equal sign and the
bases are the same, the exponents are equal. You needed to change the bases of the
powers to a common base for some questions.
Another method was to guess and check. This method was useful when the bases
could not be changed to a common base.
You can use substitution or graphing to check or verify your solution.
You explored how the compound interest formula could be used to determine how long
you would need to invest money in order to have the money grow to a certain amount.
In future lessons you will learn an algebraic method for solving exponential equations
when the bases do not share a common base.
Math 30-1: Module 6 Assignment
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Lesson 3: Understanding Logarithms
( )
1. Rewrite the logarithmic equation logR 1 = - H in exponential form with positive
M
exponents.
2. Evaluate each logarithm and explain the process used.
a.
log4 1024
æ ö
b. log2 çç 1 ÷
è8 ÷
ø
c. log3
27
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3. Determine the value of a in each of the following equations.
a. log6 a = 3
b. loga 16 = 2
3
4. In the beginning of this lesson the idea that the growth of a tree could be modelled using a
logarithmic function was introduced. A possible logarithmic function that could model the
growth of a tree is h  4  log3 t, where h is the height of the tree in metres and t is the
time in years. Estimate the approximate height of the tree to the nearest tenth of a metre
after 10 yr. Explain the reasoning for your estimation.
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A , where A is the
5. The formula for the Richter magnitude, M, of an earthquake is M = log
Ao
amplitude of the earthquake and A0 is the amplitude of a standard earthquake with
magnitude of 0.
In 2011, an earthquake occurred off the Pacific coast of the Tōhoku region of Japan. This
earthquake, which measured 9.0 on the Richter scale, is one of the five most powerful
earthquakes in the world since modern record-keeping began in 1900.
By comparison, one of the largest earthquakes to hit Alberta was a magnitude 4.7
earthquake. This earthquake occurred in 1984, on the border between British Columbia
and Alberta, just north of Waterton Lakes National Park. Determine how many times larger
in amplitude the Tōhoku earthquake was compared to the Alberta quake. Answer to the
nearest ten thousand.
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LESSON 3 SUMMARY
In this lesson you explored the concept of logarithms. The inverse of an exponential
function, y = cx, c > 0, c ≠ 1, is a logarithmic function, y = logc x, c > 0, c ≠ 1. A
logarithm of a number is the exponent by which a given base must be raised to produce
that number. You can change between exponential and logarithmic forms.
Exponential
Form
Logarithmic
Form
x = cy
y = logc x
The logarithmic function is the reflection of the exponential function in the line y = x. The
characteristics of the logarithmic function y = logc x, c > 0, c ≠ 1 are as follows:
o
o
o
o
The domain is {x|x > 0, x ∈ R}.
The range is {y|y ∈ R}.
The x-intercept is 1.
The vertical asymptote is x = 0, or the y-axis.
In the next lesson you will explore the graphs of different logarithmic functions
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Lesson 4: Graphing Logarithmic Functions
1. Consider the functions f(x)  log2 x and g(x)  log2 x (x  3)  4.
a. Find the domain and range for both functions.
b. Explain how the graph g(x) can be described as a transformation of f(x).
2. The graph of f(x)  log3 x is
 vertically stretched about the x-axis by a factor of 2
 horizontally stretched about the y-axis by a factor of 3
 translated 5 units to the right and 2 units down
If the equation of the transformed image is written in the form g (x ) = a log3 éëêb (x - c )ù
ú
û+ d ,
determine the values of a, b, c, and d. Then write the equation of g(x).
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3. The slope of a beach is influenced by the local tides, wind and wave patterns, and the size
of the sand particles on the beach. In general, as the diameter in millimetres, d, of the
average size of the particles on the beach increases, the slope of the beach, s, increases
according to the formula s(d)  0.159  0.1181 log d.
Creatas/Thinkstock
a. Describe the graph of the function s(d) as compared to the graph of f (d)  log d using
transformations terminology.
b. If the slope of a particular beach is 0.005 (about a 1-m vertical climb for every 200 m in
the horizontal direction), determine the average diameter (in millimetres) of a grain of
sand on the beach. Round your answer to 4 decimal places.
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LESSON 4 SUMMARY
In this lesson you looked at graphing logarithmic functions and how transformations can
be applied to logarithmic functions.
When a logarithmic function is in the form
the parameters a, b,
h, and k correspond to the following transformations of the graph of y = logc x:
Parameter
Value > 0
Value < 0
a
o
vertical stretch by a
factor of |a|
o
vertical stretch by a
factor of |a| and
reflection in the x-axis
b
o
horizontal stretch of
o
horizontal stretch by a
graph by a factor of
factor of
and
reflection in the y-axis
h
o
translation of h units to
the right
o
translation of |h| units
to the left
k
o
translation of k units up
o
translation of |k| units
down
In the next lesson you will continue to work with logarithms. You will explore how to
work with multiple logarithms in an expression.
Math 30-1: Module 6 Assignment
Lesson 5: Laws of Logarithms
1. Use the laws of logarithms to evaluate each expression.
a. log10 106 - log10 109
b. log3 108 – log3 4
c.
1 log 9 - log 12 + 2log 6
3
3
2 3
d. 3log2 6 - 3log2 3
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2. Expand each of the following expressions:
a. log5 (2x)
b. log8 (x3)
æ 4ö
÷
c. log2 ççç 8 x ÷
÷
÷
çè 5 ø
3. Write each expression as a single logarithm in simplest form. State any restrictions on the
variable.
a. log2 (x2  16)  log2 (2x  8)
Math 30-1: Module 6 Assignment
b. log5 (x2  16)  log5 (x2  2x – 8)
c. 3 log8 (x  3)  log8 (x2  x  6)
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LESSON 5 SUMMARY
In this lesson you explored the laws of logarithms. You used numeric values to
determine what the laws should be and, then, looked at how the laws could be derived.
You studied the following laws:
o
product law of logarithms:
logb (M × N) = logb M + logb N
o
quotient law of logarithms:
o
power law of logarithms:
logb (Mn) = nlogb M
The laws of logarithms can be used to evaluate logarithmic expressions as well as
expand or simplify logarithmic expressions.
You also saw how logarithmic scales are used. You solved problems involving the
decibel scale and the pH scale.
Using the laws of logarithms to expand or simplify logarithmic expressions will be helpful
in the next lesson. In Lesson 6 you will solve logarithmic equations.
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Lesson 6: Logarithmic and Exponential Equations
1. Provide worked examples for the last row of the concept chart shown.
1
1
Source: Pre-Calculus 12. Whitby, ON: McGraw-Hill Ryerson, 2011. Reproduced with permission.
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2. The equation log10 y = n log10 x + log10 k can be used to describe the straight-line graph
shown. Determine the values of k and n.
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3. How long, to the nearest tenth of a year, will it take a $5000 investment to double if the
money is invested at 7% interest compounded quarterly? How long will it take to triple the
investment?
4. There is a relationship between mortgage amount, number of payments, amount of each
payment, how often each payment is made, and the interest rate. The following formula
illustrates the relationship:
PV =




- nù
é
R ëê1- (1 + i ) û
ú
, where
i
PV  present value
i  interest rate per compounding period, as a decimal
R  payment amount
n  number of payments
Suppose a bank offers you a 3.4% interest rate, compounded semi-annually, on a
mortgage amount of $173 112. You pay $1000/mo, or $6000 semi-annually. How many
semi-annual payments would you have to make to pay off the mortgage? How long, in
years, would it take to pay off the mortgage?
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LESSON 6 SUMMARY
In this lesson you explored how to solve logarithmic and exponential equations.
To solve a logarithmic equation, apply the laws of logarithms to express the left side and
the right side of the equation as a single logarithm. You can then use the property if logc
A = logc B, then A = B, where c, A, B > 0 and c ≠ 1.
To solve an exponential equation where the bases are not powers of one another, you
can use the property if A = B, then logc A = logc B, where c, A, B > 0 and c ≠ 1.
Check that solutions satisfy the original equation. Remember that the logarithm of a
number ≤ 0 is undefined.
There are real-world situations that can be modeled with an exponential or logarithmic
equation. Some applications involve finances, such as investments and loans. Other
applications can be found in science, such as radioactive decay. Many exponential
equations can be expressed in the form
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MODULE 6 – EXPONENTS & LOGARITHMS SUMMATIVE
ASSIGNMENT
Complete the following questions from your text book. Show steps completely and clearly, as
marks are assigned for mathematical literacy and communication. Always use graph paper,
rulers, and pencils as necessary. Attach securely to this booklet before you hand everything in.
Module 6 is now complete. Once you have received your corrected work, review your
instructor’s comments and prepare for your module six test.
Text: Pre-Calculus 12 - Chapters 7 and 8
Chapter 7 – Exponential Functions
Section 7.1: Page 342 to 345 #7, 10, 11
Section 7.2: Page 354 to 357 #6, 7, 11, 12
Section 7.3: Page 364 to 365 #2, 3, 4, 14
Chapter 8 – Logarithmic Functions
Section 8.1: Page 380 to 382 #2all, 3all, 4all, 12all, 13all, 14a, 15, 20
Section 8.2: Page 389 to 391 #4all, 8, 16
Section 8.3: Page 400 to 403 #1a,c, 2a,c,d, 3a,d, 7a,b,e, 8a,c, 10a, 11a,c,
Section 8.4: Page 412 to 415 #1a,c, 2a,d, 4a,b,5a,c, 7a,d, 8a,d,e 13