Section 7.2 Page 370 to 377

Techniques for Solving Exponential Equations

7.2

Radioactive decay is the process by which an unstable element transforms into a different element, typically releasing energy as it does so. The half-life of a radioactive substance is the time required for a sample of the material to decay to half its initial amount.

The amount, A, in milligrams, of a radioactive material remaining as a function of time, t, in years,

can be modelled by the equation A(t) � A0 ( 1 _ 2 )

t _ h

,

where A0 is the initial amount, in milligrams, and h is the half-life, in years.

In practice, A, A0, and t can be measured for an element. Once these values are known, they can be used to determine the half-life.

Example 1 Radioactive Decay

Polonium-218 is a radioactive substance that spontaneously decays into lead-214. One minute after a 100-mg sample of polonium-218 is placed into a nuclear chamber, only 80 mg remains.

a) Determine the half-life of polonium-218.

b) Graph the decay function for polonium-218.

Solution

a) Substitute A0 � 100, t � 1, and A(1) � 80 into the decay equation and solve for h.

A(t) � A0 ( 1 _ 2 )

t _ h

80 � 100 ( 1 _ 2 )

1 _ h

0.8 � 0. 5 1 _ h

Divide both sides by 100, and write

1 _

2 as 0.5.

log 0.8 � log 0. 5 1 _ h

Take the common logarithm of both sides.

log 0.8 � 1 _ h

log 0.5 Apply the power law of logarithms.

h log 0.8 � log 0.5 Multiply both sides by h.

C O N N E C T I O N S

British physicist Ernest Rutherford

(1871–1937) and others

discovered that radioactivity

was very complicated. Diff erent

types of decay can occur, but

Rutherford was the fi rst to realize

that they all occur with the

same approximately exponential

formula, the half-life equation.

370 MHR • Advanced Functions • Chapter 7

h � log 0.5

__ log 0.8

Solve for h.

h � 3.1

The half-life of polonium-218 is approximately 3.1 min.

b) Substitute the initial amount and the half-life into the decay equation to graph the function.

A(t) � A0 ( 1 _ 2 )

t _ h

A(t) � 100 ( 1 _ 2 )

t _ 3.1

A

0 � 100 and h � 3.1.

Use graphing technology. Choose appropriate window settings to view the graph.

Example 2 Powers With Diff erent Bases

Solve the equation 42x � 1 � 3x � 2.

Solution

Method 1: Apply Algebraic Reasoning

Take the logarithm of both sides and apply the power law of logarithms to remove the variables from the exponents.

42x � 1 � 3x � 2

log 42x � 1 � log 3x � 2

(2x � 1) log 4 � (x � 2) log 3 Apply the power law of logarithms.

2x log 4 � log 4 � x log 3 � 2 log 3 Apply the distributive property.

2x log 4 � x log 3 � 2 log 3 � log 4 Collect variable terms on one side of the equation.

x(2 log 4 � log 3) � 2 log 3 � log 4 Factor x on the left side.

x � 2 log 3 � log 4

___ 2 log 4 � log 3

Solve for x.

This is an exact expression for the root of the equation. Use a calculator to determine an approximate value for this expression.

The solution to this equation is x � 2.14.

7.2 Techniques for Solving Exponential Equations • MHR 371

Method 2: Use a Graphing Calculator

Enter the left side of the equation as one function and the right side as another function. Identify where the graphs of these functions intersect. Experiment with the window settings in order to view the point of intersection.

Use the Intersect operation to determine the coordinates of the point of intersection.

The graph shows that the expressions 42x � 1 and 3x � 2 are equal when x � 2.14, which confi rms the result derived algebraically above.

Example 3 Apply the Quadratic Formula

Solve the equation 2x � 2�x � 4 algebraically, and then check the solution using a graphing calculator.

Solution

There is no immediately obvious method of solving this equation. Notice that if you multiply both sides of the equation by 2x, then a quadratic equation is obtained in terms of 2x.

2x � 2�x � 4 2x(2x � 2�x) � 2x(4) 2x(2x) � 2x(2�x) � 2x(4) Apply the distributive property.

22x � 20 � 2x(4) Apply the product rule for exponents.

22x � 1 � 2x(4) Recall that a0 � 1.

This is a quadratic equation in which the variable is 2x. To see this more clearly, apply the power of a power law to the term 22x.

(2x)2 � 1 � 2x(4)

(2x)2 � 4(2x) � 1 � 0 Write in standard form: az 2 � bz � c � 0, where z � 2x.

The coeffi cients of the quadratic expression on the left side are a � 1, b � �4, and c � �1. This expression is not factorable. Apply the quadratic formula to determine the roots.

Connecting

Problem Solving

Reasoning and Proving

Reflecting

Selecting ToolsRepresenting

Communicating


Method 1: Solve for 2x Directly

2x � �b � √ �� b2 � 4ac ____

2a

� �(�4) � √ �� (�4)2 � 4(1)(�1)

______ 2(1)

� 4 � √ �� 20 __ 2

� 4 � 2 √ � 5 __ 2 Simplify the radical.

� 2 � √ � 5 Simplify the rational expression.

Method 2: Let k � 2x and Solve for k

k2 � 4k � 1 � 0

k � �b � √ �� b2 � 4ac ____ 2a

k � �(�4) � √ �� (�4)2 � 4(1)(�1)

______ 2(1)

k � 2 � √ � 5

Now substitute 2x � k.

2x � 2 � √ � 5

This equation appears to give two solutions. Explore each case.

Case 1

2x � 2 � √ � 5

log 2x � log (2 � √ � 5 ) Take the common logarithm of both sides.

x log 2 � log (2 � √ � 5 ) Apply the power law of logarithms.

x � log (2 � √ � 5 )

___ log 2

Divide both sides by log 2.

Use a calculator to evaluate this expression.

One root of this equation is approximately 2.08.

Case 2

2x � 2 � √ � 5

Note that 2 � √ � 5 � 0. A power is always a positive number, so it cannot be equal to a negative number. Thus, this root is extraneous.

Use a graphing calculator to verify these results. Use reasonable window settings to view the point of intersection.

Use the Intersect operation to determine the coordinates of the point of intersection.

This confi rms the algebraic result.There is a single solution to the equation 2x � 2�x � 4 and its value is x � 2.08.


<< >>

Example 3 illustrates an important point: when both sides of an equation are multiplied by the same non-constant factor, the possibility of obtaining extraneous roots is introduced. When this technique is used to solve an equation, the roots obtained must be checked to ensure that they are valid solutions to the original equation. Any extraneous roots should be rejected as invalid.

KEY CONCEPTS

An equation maintains balance when the common logarithm is applied to both sides.

The power law of logarithms is a useful tool for solving for a variable that appears as part of an exponent.

When a quadratic equation is obtained, methods such as factoring and applying the quadratic formula may be useful.

Some algebraic methods of solving exponential equations lead to extraneous roots, which are not valid solutions to the original equation.

Communicate Your Understanding

C1 Refer to the radioactive decay equation A(t) � A0 ( 1 _ 2 )

t _ h

.

a) Why is there a factor that contains the fraction 1 _ 2 ?

b) Explain the signifi cance of the fraction 1 _ 2 in the context of this

question.

C2 The solution to the equation 53x � 4x � 1 is given below, but in jumbled order.

x(3 log 5 � log 4) � log 4

53x � 4x � 1

3x log 5 � x log 4 � log 4

log 53x � log 4x � 1

3x log 5 � (x � 1) log 4

3x log 5 � x log 4 � log 4

x � log 4 ___

3 log 5 � log 4

a) Copy the solution into your notebook, with the steps rearranged in the correct order.

b) Write a brief explanation beside each step.

C3 Refer to Example 3. Could this equation be solved by taking the common logarithm of both sides? If so, provide a solution. If not, explain the problem with this method.

C4 a) What is an extraneous root?

b) How can extraneous roots be identifi ed?

c) Give an example of an extraneous root for an exponential equation.


A Practise

For help with questions 1 to 3, refer to Example 1.

1. Match each graph with its equation.

a)

b)

c)

d)

i) y � 100 ( 1 _ 2 )

x

ii) y � 50 ( 1 _ 2 )

x _ 4

iii) y � 50 ( 1 _ 2 )

x

iv) y � 100 ( 1 _ 2 )

x _ 4

y

x2 4 6 8

10

20

30

40

0

y

x2 4 6 8

10

20

30

40

0

y

x4 8 12 16

10

20

30

40

0

y

x4 8 12 16

10

20

30

40

0

2. Solve for t. Round answers to two decimal places.a) 2 � 1.07t b) 3 � 1.1t

c) 100 � 10(1.04)t d) 5 � 1.08t � 2

e) 0.5 � 1.2t � 1 f) 10 � ( 1 _ 4 )

3t

g) 15 � ( 1 _ 2 )

t _ 4

h) 100 � 25 ( 1 _

2 )

6 _ t

3. Use the decay equation for polonium-218,

A(t) � A0 ( 1 _ 2 )

t _ 3.1

, where A is the amount

remaining after t minutes and A0 is theinitial amount.a) How much will remain after 90 s from

an initial sample of 50 mg?b) How long will it take for this sample to

decay to 10% of its initial amount of 50 mg?c) Would your answer to part b) change if the

size of the initial sample were changed? Explain why or why not.

For help with questions 4 and 5, refer to Example 2.

4. Solve each equation. Leave answers in exact form.

a) 2x � 3x � 1

b) 5x � 2 � 4x

c) 8x � 1 � 3x � 1

d) 72x � 1 � 4x � 2

5. Use Technology Find approximate values for your answers to question 4, correct to three decimal places.

For help with questions 6 and 7, refer to Example 3.

6. Consider the equation 22x � 2x � 6 � 0.

a) Write the equation in the form az2 � bz � c � 0, where z � 2x, and then identify the coeffi cients a, b, and c.

b) Solve the equation using the quadratic formula.

c) Identify any extraneous roots.

7. Consider the equation 82x � 2(8x) � 5 � 0.

a) Write the equation as a quadratic equation of 8x, as in question 6, and then identify the coeffi cients a, b, and c.

b) Solve the equation using the quadratic formula.

c) Identify any extraneous roots.


B Connect and Apply

8. A 20-mg sample of thorium-233 decays to 17 mg after 5 min.

a) Determine the half-life of thorium-233.

b) How long will it take for this sample to decay to 1 mg?

9. A 10-mg sample of bismuth-214 decays to 9 mg in 3 min.

a) Determine the half-life of bismuth-214.

b) Graph the amount of bismuth-214 remaining as a function of time.

c) Describe how the graph would change if the half-life were

i) shorter ii) longer

Give reasons for your answers.

d) Describe how the graph would change if the initial sample size were

i) greater ii) less

Give reasons for your answers.

10. Does the equation 42x � 2(4x) � 3 � 0 have any real solutions? Explain your answer.

11. Solve and check for extraneous roots. Where necessary, round answers to 2 decimal places.

a) 32x � 7(3x) � 12

b) 5x � 3 � 52x

c) 8(3)2x � 3 � 24

d) 18 � 2(9)2x � 1

e) 3x � 1 � 56(3�x) � 0

f) 4x � 3 � 18(4�x)

12. Use Technology Check your solutions to question 11 using graphing technology.

13. Alchemy is the ancient study of turning substances into gold using chemical, or sometimes magical, means. When platinum-197 undergoes a nuclear process known as beta decay, it becomes gold-197, which is a stable isotope. The half-life of platinum-197 is 20 h.

a) How long would it take to turn 90% of a 1-kg sample of platinum-197 into gold?

b) Do you think this is a viable “get-rich-quick” scheme? Why or why not?

14. The probability, P, in percent, that a person will respond to a certain advertisement can be modelled by the equation P(n) � 1 � 2�0.05n, where n is the number of days since the advertisement began on television. After how many days is the probability greater than 20%?

15. Lana has purchased a minivan for $24 000. The value, V, in dollars, of the minivan as a function of time, t, in years, depreciates

according to the function V(t) � 24 000 ( 1 _ 2 )

t _ 3

.

How long will it take for Lana’s minivan to depreciate to

a) one quarter of its initial value?

b) 10% of its initial value?

16. Chapter Problem One important characteristic of an electric guitar is its sustain, which is related to the length of time a plucked string continues to sound before the sound dies out. One particular guitar can hold a certain note

according to the relation I(t) � I0 ( 1 _ 2 )

t _ 12

,

where I is the intensity of the note as a function of time, t, in seconds, and I0 is the initial intensity of the note when it is struck.

a) How long will it take for the intensity of this note to decay to

i) half of its initial value?

ii) 10% of its initial value?

b) Sketch a graph of the sustain function.

c) Suppose that the design of the guitar is improved so that its sustain is increased. What effect will this have on the i) equation? ii) graph?

Connecting

Problem Solving

Reasoning and Proving

Reflecting

Selecting ToolsRepresenting

Communicating

C O N N E C T I O N S

The element bismuth is used in making chemical alloys, medicines,

and transistors.


✓ Achievement Check

17. One day, a controversial video is posted on the Internet that seemingly gives concrete evidence of life on other planets. Suppose that 50 people see the video the fi rst day after it is posted and that this number doubles every day after that.

a) Write an expression to describe the number of people who have seen the video t days after it is posted.

b) One week later, a second video is posted that reveals the fi rst as a hoax. Suppose that 20 people see this video the fi rst day after it is posted and that this number triples every day after that. Write an expression to describe the number of people who have seen the second video t days after it is posted.

c) Set the two expressions from parts a) and b) equal and solve for t using tools and strategies of your choice. What does this solution mean?

d) If you solved the equation in part c) algebraically, check your answer using graphing technology. If you solved it graphically, check your answer using algebraic reasoning.

C Extend and Challenge

18. When platinum-197 undergoes beta decay, it becomes gold-197. The half-life of platinum-197 is 20 h.

a) Develop an equation for the amount of gold-197 produced as a function of time for the given scenario.

b) Find the domain and the range of this function.

19. a) Perform some research on the carbon fusion cycle. Describe the steps of the cycle and how energy is produced.

b) Pose and solve two problems related to the carbon fusion cycle that involve exponential equations.

20. Math Contest If 4x � 4x � 1 � 24, what is the value of (2x)x?

A 5 √ � 5 B 25 C 125 D 25 √ � 5

21. Math Contest If 2a � 5 and 2b � 3, what is the value of log3 10?

A b � 1 __ a

B b � 1 __ a � 1

C a � 1 __ b

D a _ b

E b _ a

22. Math Contest Given a function h(x) that satisfi es the equation h(2x � 1) � 2h(x) � 1 for all x, and h(0) � 2, what is the value of h(3)?


Section 7.2 Page 370 to 377

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Transcript of Section 7.2 Page 370 to 377