Chapter 6 Prerequisite Skillspjk.scripts.mit.edu/lab/mhf/MHR(2008)_MHF4U_Chapter_6.pdf · ... log 64 34 = b) y = log 30 . 3. ... log 15. 5 . b) log 0.8. 2 . 6. The age of a bone can

Name: _______________________________ Date: ________________________

Chapter 6 Prerequisite Skills …BLM 6–1. . Graph an Exponential Function 1. Consider the function y = 2(3x). a) Sketch a graph of the function. b) State the domain, the range, and the

equation of the asymptote. 2. The value of a car depreciates according

to the formula , where V is the value of the car, in thousands of dollars, and t is time, in years.

( )23 0.8 tV =

a) Calculate the original value of the car.

b) Use the formula to calculate the value of the car after 3 years.

Apply the Exponential Laws 3. Simplify. Express your answers using

only positive exponents.

a) 6 2

3p p

p

−

b) ( ) ( )2 24 3n n−

×

c) ( ) ( )3 223 3x x÷ d)

4 6

2 86416

m nm n−

4. Simplify first, then evaluate. Avoid using

a calculator. a) ( )( )20 183 3−

b)

4 3

42

2 25 5

25

⎛ ⎞⎜ ⎟⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎜ ⎟

⎜ ⎟⎡ ⎤⎛ ⎞⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦⎝ ⎠

c) 3 0

3 02 22 2

−

−

+−

Graph an Inverse 5. a) Graph the function . ( ) ( )23f x x= + b) State the domain and the range of f. c) Graph ( )1y f x−= by reflecting the

graph of f in the line y = x. d) State the domain and range of 1f − . e) Explain whether 1f − is a function or

not.

6. Repeat question 5 for the function whose graph is shown below.

Apply Transformations to Functions 7. Identify the transformations required to

transform f onto g in each. a)

b)

8. Consider the function y = 3x. a) List the transformations, in the proper order,

required to transform the graph of y = 3x into

the graph of ( ) ( ) ( )21 33

xf x − += .

b) Use your answer to part a) to sketch the graph

of ( ) ( ) ( )21 33

xf x − += .

c) Check your answer using technology.

Advanced Functions 12: Teacher’s Resource Copyright © 2008 McGraw-Hill Ryerson Limited BLM 6–1 Prerequisite Skills

Name: _______________________________ Date: ________________________

6.1 The Exponential Function and Its Inverse …BLM 6–2. . 1. a) Which of the following is an

exponential function? Explain how you know.

i) x y 1 2 2 8 3 18 4 32

ii) x y 1 5 2 25 3 125 4 625

b) Write an equation for the data that is exponential.

2. Match each graph a), b), and c) with its

inverse i), ii), or iii). Then, write an equation for each function in the left column.

a) i)

b) ii) c) iii)

3. Complete the table of key features for f(x) and its inverse. 1

2( )

x

f x = ⎛ ⎞⎜ ⎟⎝ ⎠

Inverse of f

Domain

Range

x-intercept

y-intercept

Intervals for which f(x) is positive

Intervals for which f(x) is increasing

Equation of asymptote

4. The deeper you are under water, the less sunlight

reaches you. The percent of sunlight, P, that reaches a depth d, in metres, can be modelled by the function . ( ) ( )100 0.85 dP d =

a) Sketch a graph of P for the interval [0, 10]. b) Sketch the inverse of P. c) Use your graph of the inverse of P to determine

the depth at which the percentage of sunlight is 50%. Test your value for depth in the original equation for P .

d) Use your graph of P to calculate the average rate at which the sunlight is absorbed over the first 5 m.

e) Is the instantaneous rate at which sunlight is absorbed at a depth of 5 m greater than your answer in part d)? Explain.

5. Use Technology Use The Geometer’s

Sketchpad® to construct the graph of y = bx and its inverse, where b is a parameter with least value 1.

a) As the value of b increases from 1 to 2, what happens to the rates of change of the functions?

b) Estimate the value of b for which y = bx and its inverse have only one point of intersection.

Advanced Functions 12: Teacher’s Resource Copyright © 2008 McGraw-Hill Ryerson Limited BLM 6–2 Section 6.1 Practice

Name: _______________________________ Date: ________________________

6.2 Logarithms …BLM 6–3. . 1. Rewrite each equation in logarithmic

form:

a) 35 = 243 b) 31 6216

−=

2. Rewrite each equation in exponential

form: a) b) 4log 64 3= log 30y = 3. Evaluate without using a calculator.

a) b) 8log 64 21log64

c) l d) og100 2.6log10−

4. Use a graphical method to estimate the

value of each, to one decimal place. a) b) 2log 5 2log 3 5. Evaluate, correct to two decimal places,

using a calculator: a) b) 5log 15 2log 0.8 6. The age of a bone can be determined from

the fraction of carbon-14 that remains in the bone. The age is calculated by using the formula , where A is the age in years and R is the fraction of carbon-14 remaining.

(19 000 logA R= − )

a) How old is a bone that has only 34

of

its original carbon-14? b) How old is a bone that has only 10%

of its original carbon-14? c) Express the formula in exponential

form. d) Use your answer to part c) to

calculate the percent of carbon-14 remaining in a bone from an animal that died 100 years ago.

7. In the Key Concepts of this section, it is

stated that “the value of logb x is equal to the exponent to which the base, b is raised to produce x.” Use this definition to explain why . 20

4log 4 20=

8. The pH scale used to specify the acidity of a solution is given by the formula

( )logpH = − C , where C is the concentration of hydronium ions in a solution in moles per litre.

a) A strong acid has a concentration of 32.6 10−× mol/L. What is its pH?

b) Express the formula in exponential form.

c) Use your answer to part b) to determine the concentration of hydronium ions in a neutral solution (pH = 7).

9. Use a calculator. a) Evaluate the following, correct to two

decimal places. , log3.16 ( )2log 3.16 , , ( )3log 3.16

( )4log 3.16

Find a pattern in your answers. b) Use the pattern you found in part a)

to calculate ( )10log 3.16 and

( )2.5log 3.16 . Check your answers

with a calculator. c) Use the pattern to calculate ( )50log 100 ,

( )52log 32 , and log 1000

10. Use a calculator. a) Evaluate the following, accurate to

one decimal place: , , and . Find a pattern in your

answers.

log 4 log 20log80

b) Use the pattern you found in part a) to calculate l and og(80 4)×

)log(80 20× . Check your answers with a calculator.

c) Use the pattern to calculate ( )2log 8 32× and ( )5log 25 5 .


Name: _______________________________ Date: ________________________

6.3 Transformations of Logarithmic Functions …BLM 6–4. . (page 1) 1. Match the following equations with one

of the graphs below. ( )log 2y x= ( )log 2y x= +

( )2logy x= ( )log 2y x= + a)

b)

c)

d)

2. Sketch a graph of each function. For

each, state the domain and the range. a) b) logy x= log 2y x= −

c) d) 2logy x= − ( )log 2y x= −

e) 1log2

y x⎛ ⎞= ⎜ ⎟⎝ ⎠

f) ( )log 2y x= −

3. The growth of a $1000 investment at an interest rate of 6% per year compounded annually can be modelled by the function ( ) 40log 120n A A= − , where n is the

number of years needed to grow to A dollars.

a) Use the formula to calculate the number of years needed for the investment to

i) double to $2000 ii) triple to $3000 b) Sketch a graph of n versus A for

0 3000A≤ ≤ . Then, use your graph to estimate the value of the investment after 8 years.

c) In real life there must be a restriction on the domain of this function. What is this restriction? Explain.

4. The following graphs are transformations

of the graph of logy x= . Write a possible equation for each.

a)

b)

5. Sketch a graph of each function. a) ( ) ( )2log 3f x x= − +

b) ( ) ( )log 2 3 4f x x= ⎡ + ⎤ −⎣ ⎦

c) ( ) ( )( )3log 3f x x= − +

d) ( ) ( )1 log 2 62

f x x= +


Name: _______________________________ Date: ________________________

6. Describe how the graph of each function

can be obtained using transformations of the graph of . logy x=

a) ( )log 2 3y x= − + b) ( )3log 4y x= − +7. Use transformations to explain why

( )logy x= − − and are inverses of each other.

logy = x

…BLM 6–4. . (page 2) 8. Use Technology a) Compare the graphs of each pair of

functions. i) log 1y x= + and log10y x=

ii) log 2y x= + and 2log10y x=

iii) log 3y x= + and 3log10y x= b) Use the pattern from part a) to graph

, without using technology.

4log10y = x


Name: _______________________________ Date: ________________________

6.4 Power Law of Logarithms …BLM 6–5. . 1. Use the power law to evaluate without

using a calculator. a) b) 7

3log 27 52log 4−

c) 35log 25 d) 8log 2

2. Solve for n, correct to three decimal

places. a) 5 2 .5n=

b) ( )2000 500 1.045 n= 3. A child is sitting on a moving swing. The

horizontal displacement of the swing away from the vertical is modelled by the function , where a is the displacement, in metres, and n is the number of swings since the child’s father stopped pushing.

( )2.7 0.9 na =

a) What is the displacement after the sixth swing?

b) After how many swings is the displacement less than 1 m?

4. Use a calculator to evaluate, to one

decimal place. a) b) 9log 12 0.25log 52 5. Write as a single logarithm, then evaluate

without a calculator.

a) log16log 4

b)

8log272log3

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

6. Solve, to two decimal places. a) lo g 4 7x =

b) 312 log 4m=

7. Suppose the temperature of the atmosphere is increasing by 1% each year. The temperature T can be modelled by the function 1.011758.27 logy T= + , where y is the year and T is the average temperature of the atmosphere, in degrees Celsius.

a) Verify that the average temperature was 12°C in 2008.

b) In what year will the average temperature be 15°C?

8. A particular radioactive isotope has a

half-life of 400 years The time required for only A% of the original amount to remain is modelled by the function

2400log100

At ⎛= − ⎜⎝ ⎠

⎞⎟ , where t is the time ,

in years. a) After how many years will only 25%

of the original amount of radioactive isotope remain?

b) After 20 years, what percent of the original amount of radioactive isotope remains?

9. To understand why is not defined

when b = 1, do the following. log 5b

a) Let 1log 5x = , convert to exponential form, and try to solve for x.

b) Use the power law to express as a ratio of common logarithms.

1log 5

In both cases, explain why . 1b ≠ 10. Use Technology Graph as a

varies from 2 to 10. logay x=

a) What kind of transformation appears to be occurring as a result of changing the value of a?

b) Write logay x= in terms of common logarithms. Explain how this way of writing the function relates to your answer to part a).


Name: _______________________________ Date: ________________________

6.5 Making Connections: Logarithmic Scales …BLM 6–6. . in the Physical Sciences 1. Determine the pH, correct to one

decimal place, of a solution with each hydronium ion concentration.

a) 0.000 316 mol/L b) 7.9 × 10−9 mol/L

2. Calculate the hydronium ion

concentration, correct to two decimal places, if the pH of a solution is

a) 2.2 b) 11.6 3. Use the sound level scale given on

page 351 of the text. a) How many times as intense is a

normal conversation compared to a whisper?

b) How many times as intense is normal city traffic compared to a shout?

4. The intensity of sound in a library is

estimated to be one thousandth that of normal conversation. What is the decibel rating for the library?

5. How many times as intense is an

earthquake with a magnitude of 7.2 than an earthquake with a magnitude of 5.6?

6. If an earthquake is 390 times as intense

as an earthquake with a magnitude of 4.2 on the Richter scale, what is the magnitude of the more intense earthquake?

7. The absolute magnitude of star A is –4.5

and that of star B is 0.2. a) How many times as bright is star A

than star B, to the nearest unit? b) If the apparent magnitudes of two

stars are –2.5 and 1.3, respectively, which star is closer to Earth? Justify your answer.

8. The amount of power per unit area carried by sound at the threshold of hearing (0 dB) is 10−12 W/m2. Calculate the power per unit area carried by sound from each source.

a) a normal conversation (60 dB) b) a chain saw operating at a distance of

1 m (117 dB) 9. If the Richter scale is altered so that it

compares energy released instead of intensity , the definition of the scale

becomes 22 1 31

1log EM M

E⎛ ⎞

− = ⎜ ⎟⎝ ⎠

.

a) If the magnitude of an earthquake is increased by 1 on the scale, by what factor is the energy released multiplied?

b) What is the magnitude of an earthquake that releases 200 times the energy of an earthquake with magnitude 4.5?

c) How many times greater is the energy released from an earthquake with magnitude 9.0 than that of an earthquake with magnitude 6.5?


Name: _______________________________ Date: ________________________

Chapter 6 Review …BLM 6–8. . (page 1) 6.1 The Exponential Function and Its Inverse

1. a) Graph the function ( ) 12

x

f x ⎛ ⎞= ⎜ ⎟⎝ ⎠

.

Identify the key features of the graph (domain, range, intercepts, intervals for which the function is positive and intervals for which it is negative, intervals over which the function is increasing and intervals for which it is decreasing, equation of the asymptote).

b) Graph 1f − on the same grid as f by reflecting the graph of f in the line y = x.

c) Identify the key features of 1f − . 6.2 Logarithms 2. Rewrite each equation in logarithmic

form. a) 54 = 525 b) 4x = 12 c) y = 123 3. Rewrite each equation in exponential

form. a)

b) log8x =

54 log x=logb=

c) 7 200

4. Evaluate without using a calculator.

a) 1log1000⎛⎜⎝ ⎠

⎞⎟ b) 4log 64

c) d) 2log 0.25 log10002 5. a) What is the value of ? Justify

your answer.

log 6xx

b) Is your answer to part a) true for all real values of x? Explain.

6.3 Transformations of Logarithmic Functions

6. a) A graph is produced by applying the following transformations, in order, to the graph of logy x= . • reflection in the x-axis • horizontal stretch by a factor of 2 • horizontal translation, left 5 units • vertical translation, down 3 units

b) If steps 2) and 3) of the transformations were interchanged, what would the equation of the graph be?

7. Sketch the graph of ( )4log 5 2y x= − + −

by hand. Then, check your answer using graphing technology.

8. Determine an equation for the graph

shown.

6.4 Power Law of Logarithms 9. Evaluate. Avoid using a calculator.

a) 56

4log 64⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

b) 51log

100

10. Solve for x. Round answers to two

decimal places. a) 4log 25x = b) 3x = 12 c) 7.615 x= d) ( )83000 1500 1 x= +

Advanced Functions 12: Teacher’s Resource Copyright © 2008 McGraw-Hill Ryerson Limited BLM 6–8 Chapter 6 Review

Name: _______________________________ Date: ________________________

11. The maximum intensity of a signal in a

feedback loop is given by the equation , where I is intensity

in millivolts and t is time in seconds. ( ) ( )20.65 1.25 tI t =

a) What was the original intensity of the signal?

b) In how much time will the intensity be 100 mV, to the nearest tenth of a second?

c) What will be the intensity of the signal after 3.8 s, to the nearest hundredth of a millivolt?

6.5 Making Connections: Logarithmic

Scales in the Physical Sciences 12. The magnitude, M, of an earthquake on

the Richter scale is given by

0log IM

I⎛ ⎞

= ⎜ ⎟⎝ ⎠

a) The magnitudes of two earthquakes are 4.7 and 7.1. How many times as intense was the stronger earthquake than the less severe one?

b) An earthquake is detected that is 450 times as intense as an earthquake with a magnitude of 5.2. What is the magnitude of the new earthquake?

…BLM 6–8. . (page 2) 13. The pH of a solution is calculated by

using the formula pH log H +⎡ ⎤= − ⎣ ⎦ ,

where H +⎡ ⎤⎣ ⎦ is the concentration of the

hydronium ions. a) If the pH of a solution is 4.2, what is

the concentration of the hydronium ions?

b) A strong acid has a pH less than 3. If the concentration of the hydronium ions is 41.6 10−× in a particular solution, is the solution a strong acid? Explain.

Advanced Functions 12: Teacher’s Resource Copyright © 2008 McGraw-Hill Ryerson Limited BLM 6–8 Chapter 6 Review

Name: _______________________________ Date: ________________________

Chapter 6 Test …BLM 6–10. . (page 1)

1. The graph of 32xy += is shown.

a) List the key features of the function (domain, range, intercepts, equation(s) of asymptote(s), intervals for which the function is positive and intervals for which it is negative, intervals for which the function is increasing and intervals for which it is decreasing)

b) Graph y = x and the inverse of the 32xy += on the same set of axes.

c) State the equation of the inverse. 2. a) Sketch the graph of

( )32log 4y x= − + b) State the domain, the range, and the

equation for the asymptote of . ( )32log 4y x= − +

3. Determine an equation for the graph

shown.

4. Evaluate without using a calculator. a) b) log1000 4log 256 c) 4

5log 125 d) 9log 27 5. Solve for x, correct to three decimal

places. a) log356x = b) 4log 40x =

c) 17 = 5x d) ( )2 19 40 2 x+= 6. An investment earns interest

compounded annually for 12 years. In that time, its value grows from $2500 to $7100. What was the interest rate, to the nearest tenth of a percent? Use the formula ( )1 nA P i= + .

7. The decibel scale is defined by

22 1

1

10logII

β β− =⎛ ⎞⎜ ⎟⎝ ⎠

.

a) The intensity of a sound at the threshold of hearing (0 dB) is 10−12 W/m2. What is the intensity of a 50 dB sound?

b) How many times as intense is an 85 dB sound than a 50 dB sound?

c) A noise is 400 times as intense as a 60 dB sound. What is the decibel rating of this noise?

8. The amount of radioactivity that gets

through a barrier is modelled by the

function ( ) ( )3100 0.45x

A x = , where A is the percent of radioactivity that gets through and x is the thickness of the barrier in centimetres. If 95% of the radioactivity must be stopped, how thick must the barrier be, to the nearest tenth of a centimetre?

Advanced Functions 12: Teacher’s Resource Copyright © 2008 McGraw-Hill Ryerson Limited BLM 6–10 Chapter 6 Test

Name: _______________________________ Date: ________________________

9. The magnitude of a star’s brightness is

given by the formula

12 1

2log bm m

b⎛ ⎞

− = ⎜ ⎟⎝ ⎠

, where m is the

apparent magnitude of the star (how bright it appears in the sky) and b is the brightness of the star (how much light the star actually gives off).

a) If the magnitudes of two stars are 0.5 and 3.1, how much brighter is one star than the other?

b) What is the magnitude of a star that is 2.5 times brighter than a star of magnitude 0?

…BLM 6–10. . (page 2)

Advanced Functions 12: Teacher’s Resource Copyright © 2008 McGraw-Hill Ryerson Limited BLM 6–10 Chapter 6 Test

Chapter 6 Practice Masters Answers …BLM 6–12. . (page 1)

Prerequisite Skills 1. a)

b) { } { }, , 0x y y∈ ∈ > , y = 0

2. a) $23 000 b) $11 776 3. a) p b) n2

c) 3x4 d) 6

24mn

4. a) 9 b) 52

c) 97

−

5. a)

b) { } { }, , 0x R y R y∈ ∈ ≥ c)

d) { } { }, 0 , x x y∈ ≥ ∈ e) No, the original function is not 1 to 1,

so the inverse is not a function 6. b) { } { }, 3 2 , ,1 4x x y y∈ − ≤ ≤ ∈ ≤ ≤

c)

d) { } { },1 4 , , 3 2x x y y∈ ≤ ≤ ∈ − ≤ ≤ e) Yes, the original function is 1:1, so the

inverse is a function 7. a) vertical stretch factor 2, translation left

3 units and down 4 units

b) reflection in x-axis, vertical stretch factor 2, translation right 2 units

8. a) vertical compression factor 13

,

reflection in y-axis, translation left 2 units

b)

6.1 The Exponential Function and Its Inverse 1. a) Data set b) is exponential. Successive

terms have constant ratios. b) 5xy =

2. a) iii, 3xy = b) i, 5xy =

c) ii, 13

x

y ⎛ ⎞= ⎜ ⎟⎝ ⎠

3. 1

2( )

x

f x = ⎛ ⎞⎜ ⎟⎝ ⎠

Inverse of f

Domain { }x∈ { }, 0x x∈ >

Range { }, 0y y∈ > { }y∈

x-intercept none 1 y-intercept 1 none Intervals for which f(x) is positive

( )+∞∞− , ( )0,1

Intervals for which f(x) is increasing

none none

Equation of asymptote y = 0 x = 0

4. a)

Advanced Functions 12: Teacher’s Resource Copyright © 2008 McGraw-Hill Ryerson Limited BLM 6–12 Chapter 6 Practice Masters Answers


b)

c) 4.3 m d) –11.13%/m e) –7.21 %/m; greater, magnitude of

slope is increasing as d increases 5. a) increases b) 1.44 6.2 Logarithms

1. a) b) 3log 243 5= 61log 3

216= −

2. a) 43 = 64 b) 10 30y =3. a) 2 b) –6 c) 2 d) –2.6 4. a) 2.3 b) 1.6 5. a) 1.68 b) –0.32 6. a) 2374 years b) 19 000 years

c) 19 00010A

R−

= d) 99% 7. Answers may vary. 8. a) 2.6 b) c) 10−7 mol/L 10 pHC −=

9. a) 0.50, 1.00, 1.50, 2.00, log(3.16 )2

k k=

b) 5.00, 1.25 c) 100, 25, 1.5 10. a) 0.6, 1.3, 1.9,

log log log( )m n m+ = + n

x+

}

b) 2.5, 3.2 c) 8, 2.5 6.3 Transformations of Logarithmic Functions 1. a) b) ( )2logy = ( )log 2y x= +

c) d) ( )log 2y x= ( )log 2y x=2. a) { , 0}, {x x y∈ > ∈

b){ , 0}, {x x y∈ > ∈ }

c) { , 0}, {x x y }∈ > ∈

d) { , 0}, {x x y }∈ < ∈

e) { , 0}, {x x y }∈ > ∈

f) { , 2}, {x x y }∈ > ∈

3. a) i) 12 years ii) 19 years

b) approximately $1590

c) because 1000A ≥ 0n ≥

4. a) ( )log 3y x= + b) ( )2log 5y x= − +

5. a)

b)



c)

d)

6. a) compress horizontally by a factor of

12

, translate left 3 units, reflect in the

x-axis b) reflect in the y-axis, stretch vertically

by a factor of 3, translate up 4 units 7. The function can be

obtained by reflecting y = log x in the y-axis and reflecting in the x-axis, so a point (a, b) is transformed to (−a, −b). Reflection of the function y = log x in the line y = x has the exact same effect, so the two log functions are inverses.

log( )y x= − −

8. a) each pair of graphs is identical b) translated up 4 units

6.4 Power Law of Logarithms

1. a) 21 b) –10 c) 23

d) 16

2. a) 1.756 b) 31.495 3. a) 1.4 m b) 10 4. a) 1.1 b) –2.9

5. a) , 2 b) 4log 16 23

8log27

, 3

6. a) 11.63 b) 9.51 7. b) 2030 8. a) 800 years b) 96.6% 9. a) 1 5; because x = 1b ≠ 1 5≠

b) log5log1

; because of division by 0 1b ≠

10. a) vertical compression

b) loglog

xya

= ; 1loga

acts as a vertical

compression factor, therefore as a

increases, 1loga

decreases

6.5 Making Connections: Logarithmic Scales in the Physical Sciences 1. a) 3.5 b) 8.1 2. a) 36.31 10−× mol/L

b) 122.51 10−× mol/L 3. a) 1000 b) 3.2 4. 30 dB 5. 39.8 6. 6.8 7. a) 50119 b) star B 8. a) 610− W/m2 b) 0.5 W/m2 9. a) 31 b) 6.0 c) 5351 Chapter 6 Review 1. a) { } { }, ,x y y 0∈ ∈ > , function is

always positive and decreasing, y = 0, x-intercept: none, y-intercept: 1

b)

c) { } { }, 0 ,x x y∈ > ∈

(0, 1x∈, function

positive when , negative when

)( )1, x∈ +∞ , function always

decreasing, x = 0, x-intercept: 1, y-intercept: none

2. a) 5log 625 4= b) 4log 12 x= c) 12log 3y =

3. a) 8 = 10x b) x = 54 c) 7200 b= 4. a) –3 b) 3 c) −2 d) 8 5. a) 6 b) 0, 1x x> ≠



6. a) ( )1log 5 32

y x⎡ ⎤= − + −⎢ ⎥⎣ ⎦

b) 1log 5 32

y x⎛ ⎞= − + −⎜ ⎟⎝ ⎠

7.

8. ( )5log 1y x= +

9. a) 52

b) 25

−

10. a) 2.32 b) 2.26 c) 1.43 d) 0.09 11. a) 0.65 mV b) 11.3 s c) 3.54 mV 12. a) 251 b) 7.9 13. a) mol/L 56.3 10−×

b) no, pH is 3.8 Chapter 6 Test 1. a) { } { }, ,x y y∈ ∈ > 0 , y-intercept 8,

no x-intercept, function always positive and increasing, horizontal asymptote: y = 0

b)

c) 2log 3y x= −

2. a)

b) { } { }, 4 , x x y∈ > − ∈ , x = −4

3. ( )log 5y x= − +

4. a) 3 b) 4 c) 34

d) 32

5. a) 2.551 b) 2.661 c) 1.760 d) −1.576

6. 9.1%

7. a) 710− W/m2 b) 3162 c) 86 dB

8. 11.3 cm 9. a) 398 b) –0.4


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Chapter 6 Prerequisite Skillspjk.scripts.mit.edu/lab/mhf/MHR(2008)_MHF4U_Chapter_6.pdf · ... log 64 34 = b) y = log 30 . 3. ... log 15. 5 . b) log 0.8. 2 . 6. The age of a bone can