BLM-8.1 Chapter 8 Prerequisite Skillsanurita.weebly.com/uploads/1/9/1/5/19153695/chapter_8_review.pdf · BLM 8–2 Section 8.1 Extra Practice 1. Use the definition of a logarithm

Copyright © 2012, McGraw-Hill Ryerson Limited, ISBN: 978-0-07-073887-4

Name: ___________________________________________ Date: _____________________________

BLM 8–1

Chapter 8 Prerequisite Skills 1. How are 9 and 3 27 the same? How are

they different?

2. Between which two consecutive whole numbers does the value of each root fall? Which number is it closer to?

a) 8 b) 132 c) 3 9 d) 3 100

3. Identify two rational numbers with square roots between 8 and 9.

4. Identify the base and the exponent in each of the following powers. Evaluate each power where possible.

a) 34 b) (4)5

c) x7 d) 123x

e) 131 f) 3

23

g) 1.782.1

5. Calculate.

a) 196 b) 3 4096

c) 3 9261 d) 3 3375

e) 961 f) 3 4913

6. Write each expression as powers without parentheses. Then, evaluate each expression.

a) (43)2 b) (7 × 3)4

c) 4

56

d) [(3) × 4]3

7. Determine the value of each expression.

a) 7 2(32) b) (4 3)2 (3)2

c) (2)6 ÷ 43 d) 24 22 (72 52)

8. For each table, plot the ordered pairs (x, y) and the ordered pairs (y, x). State the domain of the function and its inverse. a) b)

x y x y 2 4 6 2 1 2 4 4

0 0 1 5 1 2 2 5 2 4 5 3

9. Sketch the inverse of each graph of a relation.

a)

b)

10. Determine algebraically the equation of the inverse of each function. a) f (x) 3x b) f (x) 3x 4

c) f (x) = 4

3x

d) f (x) 3x

5

e) f (x) 1 2.5x f) f (x)12

(x 6)

11. For each of following functions, • determine the equation for the inverse, f 1(x) • sketch the graph f (x) and f 1(x) • state the domain and range of f (x) and f 1(x) a) f (x) 2x 3 b) f (x) 5 3x

c) f (x) = 12

(x 6)

d) f (x) x2 3, x 0 e) f (x) 1 x2, x 0 f) f (x) (x 3)2, x 3


Name: ___________________________________________ Date: _____________________________

BLM 8–2

Section 8.1 Extra Practice 1. Use the definition of a logarithm to evaluate

each expression.

a) log8 64 b) log 1000 c) log2 8 d) log3 81 e) log7 1 f) log4 2 g) log 0.01

h) log4 5 64

2. Express in logarithmic form.

a) 35 243

b) 1416 2

c) 22 0.25 d) 52m n 4

3. Express in exponential form.

a) log4 64 3

b) log4 8 3

2

c) log 10 000 4 d) log6 (x 2) y

4. Determine the value of x.

a) log4 x 2 b) log5 x 1 c) logx 81 4

d) log4 3

2x

5. a) Sketch the graph of the exponential function y 3x.

b) On the same grid, sketch the graph of the inverse of y 3x.

c) Explain the relationship between the characteristics of the two functions.

6. a) State the equation of the inverse

of1

3( )

x

f x .

b) Sketch the graph of the inverse. c) Identify the domain, range, and

intercepts of the inverse graph. d) Determine the equations of any

asymptotes.

7. Identify the following characteristics of the inverse graph of each function. i) the domain and range ii) the x-intercept, if it exists iii) the y-intercept, if it exists iv) the equation of the asymptote

a)

b)

8. Without using technology, estimate the value of each logarithm to one decimal place. a) log2 60 b) log3 30 c) log5 80 d) log 35

9. a) Determine the x-intercept of y log4 (x 3). b) Determine the y-intercept of y log6 x 5.

10. The point 1

16, 4

is on the graph of the

logarithmic function f (x) logc x. The point (k, 64) is on the graph of the inverse, y f 1(x). Determine the value of k.


Name: ___________________________________________ Date: _____________________________

BLM 8–3

Section 8.2 Extra Practice 1. Describe how the graph of each logarithmic

function can be obtained from the graph of y log4 x. a) y log4 (x 8) 1 b) y log4 (3x)

c) 1

2log ( 10) 9 y x

2. a) Sketch the graph of y log2 x. Then, apply, in order, the following transformations. • Stretch horizontally by a factor

of 3 about the y-axis. • Translate 5 units to the right.

b) Write the equation of the final transformed image.

3. a) Sketch the graph of y log6 x. Then, apply, in order, the following transformations. • Reflect in the x-axis. • Translate vertically 2 units down.

b) Write the equation of the final transformed image.

4. Sketch the graph of each function. a) y log3 (x 2) 7 b) y log2 ( (x 5)) 3 c) y 4 log5 (2x) 1

5. Identify the following characteristics of the graph of each function. i) the equation of the asymptote ii) the domain and range iii) the y-intercept, to one decimal place if

necessary iv) the x-intercept, to one decimal place if

necessary a) y log5 (x) 3 b) y 3 log2 (2(x 4)) c) y 4 log7 (x 2) 1

d) 1

2log ( 10)

y x

6. In each graph, the solid curve is a stretch and/or reflection of the dashed curve. Write the equation of each solid graph. a)

b)

c)

d)


Name: ___________________________________________ Date: _____________________________

BLM 8–3 (continued)

7. Describe, in order, a series of transformations that could be applied to y log3 x to obtain the graph of each function. a) y 2 log3 (5(x 4)) 7 b) y 0.2 log3 ((x 1)) 3

8. The graph of y log2 x has been transformed to y a log2 (b(x h) k. Determine the values of a, b, h, and k for each set of transformations. Write the equation of the transformed function. a) a reflection in the y-axis and a translation

5 units right and 2 units down

b) a vertical stretch by a factor of 1

2 about

the x-axis and a horizontal stretch about the y-axis by a factor of 4

c) a vertical stretch about the x-axis by a

factor of 2

5, a horizontal stretch about

the y-axis by a factor of 1

3, a reflection

in the x-axis, and a translation of 7 units left and 2 units up

9. Describe how the graph of each logarithmic function can be obtained from the graph of y log7 x. a) y 5 log2 (3x 15) 7 b) y 0.25 log2 (2 x) 5 c) 2(y 7) log2 (x 1)

10. a) Only a horizontal translation has been applied to the graph of y log4 x so that the graph of the transformed image passes through the point (6, 2). Determine the equation of the transformed image.

b) A vertical stretch is applied to the graph of y log3 x so that the graph of the transformed image passes through the point (2, 12). Determine the equation of the transformed image.


Name: ___________________________________________ Date: _____________________________

BLM 8–4

Section 8.3 Extra Practice 1. Write each expression in terms of the

individual logarithms of x, y, and z.

a) 7

2

log

x y

z

b) 3log ( )x yz

c) 53log ( )xyz

d) 23log xy z

2. Use the laws of logarithms to simplify and evaluate each expression.

a) 8 8 83 log 4 log 4 log 2

b) 2 log2 4 log2 5 log2 10

c) log5 25 5

d) 1

2log 9 log 3

3. Write each expression as a single logarithm in simplest form.

a) log4 x 2 log4 y b) log6 x 3 log6 y 4 log6 z

c) log log

4 4

x y

d) 2 3 log x log y

4. Evaluate each of the following.

a) If log5 x 25, determine the value of

5 25log

x.

b) Determine the value of logn ab2 if logn a 5 and logn b 3.

c) If log c 3, evaluate log 10c2. d) If loga x 3 and loga y 4, evaluate

21

log a xy

.

5. Simplify.

a) log 9 log 46 65

b) log 8 log 2a aa

6. If log5 9 k, write an algebraic expression in terms of k for each of the following.

a) log5 94

b) log5 45 c) log5 (81×125)

d) 5

4 9

25log

7. Write each expression as a single logarithm in simplest form. State any restrictions on the variable.

a) 3 3 324log log log x x x

b) 3 3 325 3

2log log log

x

xx x

8. In chemistry, the pH scale measures the acidity (07) or alkalinity (714) of a solution. It is a logarithmic scale in base 10. If neutral water has a pH of 7, what is the pH of a solution that is 4 times more alkaline than water?

9. If bleach has a pH of 13, how many times more alkaline is it than blood, which has a pH of 8?

10. An earthquake off the coast of Alaska measured 6.4 on the Richter scale. Another earthquake near Japan was 50 times worse. What was the Richter scale reading for the earthquake near Japan?


Name: ___________________________________________ Date: _____________________________

BLM 8–5

Section 8.4 Extra Practice 1. Solve.

a) log2 (3 2x) log2 (2 x) log2 3 b) log4 (x

2 1) log4 6 log4 5 c) 2 log (3 x) log 4 log (6 x)

2. Solve.

a) log2 x log2 (x 7) 3 b) log2 x 3 log2 (x 2) c) log2 (2 2x) log2 (1 x) 5

3. Solve. Round your answers to two decimal places.

a) 9x 51 b) 4x 3 260

c) 43 42x

4. Determine the value of x. Round your answers to two decimal places.

a) 2x 5x 1 b) 7x 4 83x c) 2(5x) 4x 1

5. The following shows how two students chose to solve log2 x log2 3 5.

Nicole’s work:

2 2

2 2

2 3

3

3

log - log 3 = 5

log = 5

log = log 32

= 32= 96

x

x

x

x

x

Joseph’s work:

2 2

2 2

5

3

3

3

log - log 3 = 5

log = log 32

2 =

32 =96 =

x

x

x

x

x

Which method of solving do you prefer and why?

6. The following shows how Samuel attempted

to solve the equation log 500

log 5 x .

log 500

log 5=

log 100 =

2 =

x

xx

Identify, describe, and correct Samuel’s errors.

7. Solve and check each solution. Round to two decimal places when necessary.

a) log (2x 3) log (x 2) 1 0 b) log5 (3x 1) log5 (x 3) 3 c) log2 (x 2) log2 x log2 3 d) log9 (x 5) 1 log9 (x 3)

8. The compound interest formula is A P(1 i)n, where A is the future amount, P is the present amount or principal, i is the interest rate per compounding period expressed as a decimal, and n is the number of compounding periods.

a) Livia inherits $5000 and invests in a guaranteed investment certificate (GIC) that earns 6% interest per year, compounded semi-annually. How long will it take for the GIC to be worth $10 000?

b) How long will it take for money invested at 3.5% interest per year, compounded semi-annually, to triple in value?

9. The population of a town changes by an exponential growth factor, b, every 4 years. If a population of 2350 grows to 7000 in 3 years, what is the value of b? Round your answer to two decimal places.

10. Light passing through murky water loses 30% of its intensity for every metre of water depth. At what depth will the light intensity be half of what it is at the surface? Round your answer to two decimal places.


Name: ___________________________________________ Date: _____________________________

BLM 8–6

Chapter 8 Study Guide This study guide is based on questions from the Chapter 8 Practice Test in the student resource.

Question I can … Help Needed Refer to

#1 sketch and determine the characteristics of the graph of y = logc x, c > 0, c ≠ 1

some none

Section 8.1 Example 3

#2 express a logarithmic function as an exponential function and vice versa

some none

Section 8.1 Link the Ideas

#3 explain the effects of the parameters a, b, h, and k in y = a logc (b(x – h)) + k on the graph of y = logc x, where c > 1

some none

Section 8.2 Examples 1, 2

#4 determine an equivalent form of a logarithmic expression using the laws of logarithms

some none


#5 determine an equivalent form of a logarithmic expression using the laws of logarithms

some none


#6 solve a problem by applying the laws of logarithms to logarithmic scales

some none


#7 solve a logarithmic equation and verify the solution some none


#8 evaluate logarithms using a variety of methods some none


#9 explain the effects of the parameters a, b, h, and k in y = a logc (b(x – h)) + k on the graph of y = logc x, where c > 1

some none


#10 sketch the graph of a logarithmic function by applying a set of transformations to the graph of y = logc x, where c > 1, and state the characteristics of the graph

some none


#11 solve a logarithmic equation and verify the solution some none


#12 solve an exponential equation in which the bases are not powers of one another

some none


#13 solve a problem that involves the application of exponential equations to loans, mortgages, and investments

some none



some none



some none


#16 solve a problem involving exponential growth or decay some none


#17 solve a problem by modelling a situation with an exponential or logarithmic equation

some none



Name: ___________________________________________ Date: _____________________________

BLM 8–7

Chapter 8 Test Multiple Choice For #1 to 6, select the best answer.

1. The graph of f (x) logb x, b > 1, is translated such that the equation of the new graph is expressed as y 2 f (x 1). The domain of the new function is A {x | x > 0, x R} B {x | x > 1, x R} C {x | x > 2, x R} D {x | x > 3, x R}

2. The x-intercept of the function f (x) log5 x 3 is A 53 B 0 C 1 D 53

3. The equation 2

1

3logy x can also be

written as

A 32x

y B 32y

x

C 23x y D 23y x

4. The range of the inverse function, f 1, of f (x) log4 x, is A { y | y > 0, y R} B { y | y < 0, y R} C { y | y ≥ 0, y R} D { y | y R}

5. A graph of the function y log3 x is transformed. The image of the point (3, 1) is (6, 3). The equation of the transformed function is A y 3 log3 (x 3) B y 3 log3 (x 3) C y 3 log3 (x 3) D y 3 log3 (x 3)

6. If log27 x y, then log9 x equals

A 3

2

y B 2

3

y

C 3y D 4y

Short Answer

7. If log3 5 x, express 3 3

1

22log 45 log 225

in terms of x.

8. Determine the value of x algebraically.

a) log4 x 3 b) 2

3log 64 x

c) 5log 255 x d) log3 (x 1)2 2 e) log2 (logx 256) 3

9. Solve for x. a) log (2x 3) log (x 2) log (2x 1) b) log (x 7) log (x 3) log (2x 1) c) 2 log2 (x 4) log2 x 1

10. The point (6, 4) lies on the graph of y logb x. Determine the value of b to the nearest tenth.

Extended Response 11. Solve the equation 5x 104, graphically

and algebraically. Round your answer to the nearest hundredth.

12. Given f (x) log3 x and g(x) log3 9x. a) Describe the transformation of f (x)

required to obtain g(x) as a stretch. b) Describe the transformation of f (x)

required to obtain g(x) as a translation. c) Determine the x-intercept of f (x). How

can the x-intercept of g(x) be determined using your answer to parts a) or b)?

13. Explain how the graph of

4log (3 1)

21

xy can be generated by

transforming the graph of y log4 x.

14. Identify the following characteristics of the graph of the function y 2 log4 (x 1) 3 a) the equation of the asymptote b) the domain and range c) the x-intercept and the y-intercept

15. An investment of $2000 pays interest at a rate of 3.5% per year. Determine the number of months required for the investment to grow to at least $3000 if interest is compounded monthly.

16. Radioactive iodine-131 has a half-life of 8.1 days. How long does it take for the level of radiation to reduce to 1% of the original level? Express your answer to the nearest tenth.


BLM 8–8

Chapter 8 BLM Answers BLM 8–1 Prerequisite Skills 1. Each is a root of a negative number. However,

only 3 27 3 can be evaluated because 9 is not a real number. 2. a) 2 and 3; closer to 3 b) 11 and 12; closer to 11 c) 2 and 3; closer to 2 d) 4 and 5; closer to 5 3. Example: any rational number between 64 and 81 4. a) base: 3; exponent: 4; 34 81 b) base: 4; exponent: 5; (4)5 1024 c) base: x; exponent: 7

d) base: 3x; exponent: 1

2

e) base: 13; exponent: 1; 131 = 13

f) base: 2

3; exponent: 3;

32

3

3.375

g) base: 1.78; exponent: 2.1; 1.782.1 3.3564 5. a) 14 b) 16 c) 21 d) 15 e) 31 f) 17 6. a) (4)6 4096 b) 214 194 481

c) 4

4

5 625

6 1296 d) 123 1728

7. a) 11 b) 58 c) 1 d) 44 8. a)

domain: { x | x = 2, 1, 0, 1, 2}

domain: { x | x = 4, 2, 0, 2, 4} b)

domain: { x | x = 6, 4, 1, 2, 5}

domain: {x | x = 2, 3, 4, 5}

9. a)

b)

10. a) 3

xf x b) 1 4

3

xf x

c) 1 3 4 f x x d) 1 3 15 f x x

e) 1 0.4(1 ) f x x f) 1 2 6 f x x

11. a) 1 3

2

x

f x

f (x)–domain: { x x R}; range: { y y R}

f 1(x)–domain: { x x R}; range: { y y R}

b) 1 5

3

x

f x



f (x)–domain: {x x R}; range: { y y R} f 1(x)–domain: {x x R}; range: { y y R}

c) f 1(x) 2x 12

f (x)–domain: {x x R}; range: { y y R} f 1(x)–domain: {x x R}; range: { y y R}

d) 1 3 f x x

f (x)–domain: {x x 0, x R}; range: { y y 3, y R}

f 1(x)–domain: {x x 3, x R}; range: { y y 0, y R}

e) 1 1 f x x

f (x)–domain: {x x 0, x R}; range: { y y 1, y R}


f) 1 3 f x x

f (x)–domain: {x x –3, x R}; range: { y y 0, y R}


BLM 8–2 Section 8.1 Extra Practice

1. a) 2 b) 3 c) 3 d) 4 e) 0 f) 1

2 g) 2 h) 3

5

2. a) log3 243 5 b) 16

1

4log 2 c) log2 0.25 2

d) log5 (n 4) 2m

3. a) 43 64 b) 3

24 8 c) 104 10 000 d) 6 y x 2

4. a) 16 b) 1

5 c) 3 d) 8

5. a), b)

c) Example: They are reflections of each other over the line y x. Each point on the graph of one function (x, y) appears as the point (y, x) on the other graph. 6. a) 1

3

logy x

b)



c) domain: {x x 0, x R}; range: { y y R}; x-intercept: (1, 0); y-intercept: none d) vertical asymptote at x = 0 7. a) domain: {x x 0, x R}; range: { y y R}; x-intercept: (1, 0); y-intercept: none; vertical asymptote at x = 0 b) domain: {x x 0, x R}; range: { y y R}; x-intercept: (1, 0); y-intercept: none; vertical asymptote at x = 0 8. a) 5.9 b) 3.1 c) 2.7 d) 1.5 9. a) (4, 0) b) no y-intercept 10. k 6

BLM 8–3 Section 8.2 Extra Practice 1. a) translation horizontally 8 units left and vertically 1 unit down b) reflection in the y-axis, stretch horizontally about

the y-axis by a factor of 1

3

c) reflection in the x-axis, stretch vertically about the

x-axis by a factor of 1

2, translation horizontally 10

units right and vertically 9 units up

2. a)

b) 2

1

3log 5

y x

3. a)

b) y log6 x 2

4. a)

b)

c)

5. a) equation of asymptote: x 0; domain: {x x 0, x R}; range: { y y R};

y-intercept: none; x-intercept: (1

125 , 0)

b) equation of asymptote: x 4; domain: {x x 4, x R}; range: { y y R}; y-intercept: none; x-intercept: (4.5, 0) c) equation of asymptote: x 2; domain: {x x 2, x R}; range: { y y R}; y-intercept: (0, 2.4); x-intercept: (1.4, 0) d) equation of asymptote: x 10; domain: {x x 10, x R}; range:{ y y R}; y-intercept: none; x-intercept: (12, 0)

6. a) 4

1

4log

y x or y = log4 x 1 b) y 3 log2 x

c) y log3 (2x) d) y = 4 log4 x 7. a) a vertical stretch about the x-axis by a factor of 2, a horizontal stretch about the y-axis by a factor of 1

5, a reflection in the x-axis, and a translation 4 units

right and 7 units up b) a vertical stretch about the x-axis by a factor of 0.2, a reflection in the y-axis, and a translation 1 unit left and 3 units down



8. a) a 1; b 1; h 5; k 2; y log2 ((x 5)) 2

b) 1

2a ; b 0.25; h 0; k 0; 2

1

2log (0.25 )y x

c) 2

5 a ; b 3; h 7; k 2;

2

2

5log 23 7 y x

9. a) a vertical stretch about the x-axis by a factor of 5, a horizontal stretch about the y-axis by a factor

of 1

3, a reflection in the y-axis, and translation 5

units right and 7 units down b) a vertical stretch about the x-axis by a factor of 0.25, a reflection in the y-axis, and translation 2 units right and 5 units up

c) a vertical stretch about the x-axis by a factor of 1

2

and translation 1 unit left and 7 units up 10. a) y log4 (x 10) b) y 19.02 log3 x BLM 8–4 Section 8.3 Extra Practice

1. a) 7 7 72 log log log x y z

b) 3 3 3

1 1

2 2log log log x y z

c) 5 5 53 log 3 log 3 log x y z

d) 2 2 2

1

3log log log x y z

2. a) log8 512 3 b) log2 8 3 c) log5 5

2.5 2.5 d) log 1 0

3. a) 4 2log

xy

b) 6 3 4log

xy z

c) 4logxy

d) 3100

log

xy

4. a) 23 b) 11 c) 7 d) 14 5. a) 25 b) 16 6. a) 4k b) 1 k c) 2k 3 d) 0.25k 2

7. a) 11

4,3log x x ≠ 0 b)

7

5

3log ,x x ≠ 0

8. 7.6 9. 100 000 times more 10. 8.1 BLM 8–5 Section 8.4 Extra Practice 1. a) no solution b) 29 c) 3

2. a) 8 b) 2 c) 3 3. a) 1.79 b) 1.01 c) 13.6 4. a) 1.76 b) 1.81 c) 9.32

5. Example: If Nicole's work is preferred it is because it uses the definition of logarithm to convert 5 into log2 32. Once this is done, the logarithm can be dropped from both sides of the equation. If Joseph's work is preferred, it is because it converts the logarithmic equation into an exponential function. 6. Example: Samuel’s error occurs in his first calculation: log 500 divided by log 5 does not equal log 100. To solve the equation correctly, Samuel should first calculate the log of 500 and then divide this value by the log of 5.

log 500

log 5

2.69897

0.69897

3.86

x

x

x

7. a) 2.59 b) 8 c) no solution d) 6 8. a) 23.4 compounding periods, so 11.7 years b) 63.3 compounding periods, so 31.7 years 9. b 4.29 10. 1.94 m BLM 8–7 Chapter 8 Test 1. B 2. A 3. D 4. A 5. A 6. A 7. x 3

8. a) 1

64 b) 512 c) 25 d) 4, 2 e) 2

9. a) 3.5 b) no solution c) 8 10. 0.6 11. 2.89

12. a) horizontal stretch by a factor of 1

9 about the y-axis

b) vertical translation 2 units up

c) x-intercept of f (x) is 1; the x-intercept of g(x) is 1

9, since

g(x) is a result of a horizontal stretch by a factor of 1

9

13. vertical stretch by a factor of 1

2 about the x-axis,

a horizontal stretch by a factor of 1

3 about the y-axis,

a horizontal translation 1

3 units right, and a vertical

translation 1 unit up



14. a) x 1 b) domain: {x x 1, x R}; range:

{ y y R} c) x 7

8 , y 3

15. 140 months 16. 53.8 days BLM U3–2 Unit 3 Test 1. A 2. A 3. D 4. B 5. C 6. D 7. a 3, k 1.5 8. 1 9. 27 10. 9 11. 1

12. a) y (3) 2x 2

b)

domain: {x x R}; range: { y y 0, y R}; no x-intercept; y-intercept 12

13. (0.83, 0.83) and (1, 1); Example: The two functions are inverses of each other. The points of intersection lie on the line y x, the line of reflection. 14. a) 6 b) 7 c) 3

15. a) y x2 1, x 0 b) 1

2( 1), 1

xx

y x

c) y 0.1(x 3), {x | x 3, x R} 16. a) P(t) 906 × 1.027t b) 2.7% c) 2015 17. a) 6.3 × 107 moles per litre b) 5.5

18. a) 12

0.0325

122500 1

t

A

b)

domain: {t t 0, t R}; range: {A A 2500, A R}; no x-intercept; y-intercept 2500 c) 22 years

Documents

BLM-8.1 Chapter 8 Prerequisite Skillsanurita.weebly.com/uploads/1/9/1/5/19153695/chapter_8_review.pdf · BLM 8–2 Section 8.1 Extra Practice 1. Use the definition of a logarithm