Chapter 7 R E V I E W

7.1 Equivalent Forms of Exponential Equations

1. Write each as a power of 4.

a) 64 b) 4 c) 1 _ 16

d) ( 3 √ � 8 )5

2. Write each as a power of 5.

a) 20 b) 0.8

3. Solve each equation. Check your answers using graphing technology.

a) 35x � 27x � 1 b) 82x � 1 � 32x � 3

7.2 Techniques for Solving Exponential

Equations

4. A 50-mg sample of cobalt-60 decays to 40 mg after 1.6 min.

a) Determine the half-life of cobalt-60.

b) How long will it take for the sample to decay to 5% of its initial amount?

5. Solve exactly.

a) 3x � 2 � 5x b) 2k � 2 � 3k � 1

6. Use Technology Refer to question 5.

a) Use a calculator to fi nd an approximate value for each solution, correct to three decimal places.

b) Use graphing technology to check your solutions.

7. Solve each equation. Check for extraneous roots.

a) 42x � 4x � 20 � 0 b) 2x � 12(2)�x � 7

8. A computer, originally purchased for $2000, loses value according to the exponential

equation V(t) � 2000 ( 1 _ 2 )

t _ h , where V is the

value, in dollars, of the computer at any time, t, in years, after purchase and h represents the half-life, in years, of the value of the computer.After 1 year, the computer has a value of approximately $1516.

a) What is the half-life of the value of the computer?

b) How long will it take for the computer to be worth 10% of its purchase price?

7.3 Product and Quotient Laws of Logarithms

9. Evaluate, using the laws of logarithms.

a) log6 8 � log6 27

b) log4 128 � log4 8

c) 2 log 2 � 2 log 5

d) 2 log 3 � log ( 25 _ 2 )

10. Write as a single logarithm.

a) log7 8 � log7 4 � log7 16

b) 2 log a � log (3b) � 1 _ 2 log c

11. Write as a sum or difference of logarithms. Simplify, if possible.

a) log (a2bc) b) log ( k _ √ � m

)

12. Simplify and state any restrictions on the variables.

a) log (2m � 6) � log (m2 � 9)

b) log (x2 � 2x � 15) � log (x2 � 7x � 12)

7.4 Techniques for Solving Logarithmic

Equations

13. Solve.

a) log (2x � 10) � 2 b) 1 � log (2x) � 0

14. Solve log2 x � log2 (x � 2) � 3. Check for extraneous roots.

15. Use Technology Check your answer to question 14 using graphing technology.

16. When you drink a cup of coffee or a glass of cola, or when you eat a chocolate bar, the percent, P, of caffeine remaining in your bloodstream is related to the elapsed time, t,

in hours, by t � 5 ( log P

__ log 0.5

) .

a) How long will it take for the amount of caffeine to drop to 20% of the amount consumed?

b) Suppose you drink a cup of coffee at 9:00 a.m. What percent of the caffeine will remain in your body at noon?

408 MHR • Advanced Functions • Chapter 7

7.5 Making Connections: Mathematical

Modelling With Exponential and

Logarithmic Equations

17. A savings bond offers interest at a rate of 6.6%, compounded semi-annually. Suppose that a $500 bond is purchased.

a) Write an equation for the value of the investment as a function of time, in years.

b) Determine the value of the investment after 5 years.

c) How long will it take for the investment to double in value?

d) Describe how the shape of the graph of this function would change if

i) a bonus of 1% of the principal were added after 3 years had passed

ii) the size of the initial investment were doubled

The annual Summer-Fest concert in Decimal Point has been gaining popularity. Word of the combination of excellent performing acts and a picturesque concert site has spread quickly, even to some of the large cities in the south.Attendance fi gures at Summer-Fest are shown in the table.

a) Create a scatter plot of the data.

b) Use regression to determine a curve of best fi t. Explain why your choice of curve makes sense for this situation.

c) How many fans do you expect to attend the

i) 2010 concert? ii) 2015 concert?

d) Integer Island has certain restrictions due to the physical limitations of the site:

• The concert venue has a capacity of 5000 people.

• Parking facilities can accommodate no more than 1800 vehicles.

Predict when the concert organizers will have to consider changing the location of Summer-Fest. Use mathematical reasoning to support your answer.

e) What other measures could be taken to keep Summer-Fest on Integer Island? Describe what you could do, as an urban planner, to preserve this popular northern tradition. Use mathematical reasoning to support your plans and to predict for how much longer Decimal Point can put off fi nding a new concert venue, without having to turn any fans away at the gate. Discuss any assumptions you must make.

Year Number of Paid Admissions

2000 4502001 512

2002 6062003 7182004 815

2005 956

2006 1092

2007 1220

CH

AP

TE

R

P R O B L E M W R A P - U P

Integer Island(home of

Summer-Fest!)

Logarithm Lake

Town ofDecimal Point

Squa

re R

oute

Exp

ress

way

Geometropolis(the Big City!)

Chapter 7 Review • MHR 409