Section 7.4 Page 387 to 392

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Techniques for SolvingLogarithmic Equations

7.4

The sensitivity to light intensity of the human eye, as well as of

certain optical equipment such as cameras, follows a logarithmic

relationship. Adjusting the size of the aperture that permits

light into the camera, called the f-stop, can compensate for poor

lighting conditions. A good understanding of the underlying

mathematics and optical physics is essential for the skilled

photographer in such situations.

You have seen that any positive number can be represented as

a power of any other positive base

a logarithm of any other positive base

For example, the number 4 can be written as

a power: a logarithm:

41 22 161_2

10log 4 log2

16 log3

81 log 10 000

Can any of these representations of numbers be useful for solving equations

that involve logarithms? If so, how?

Investigate How can you solve an equation involving logarithms?

1.Use Technology Consider the equation log (x 5) 2 log (x 1).

a) Describe a method of finding the solution using graphing technology.

b) Carry out your method and determine the solution.

2. a) Apply the power law of logarithms to the right side of the equation

in step 1.

b) Expand the squared binomial that results on the right side.

3. R e f l e ct

a) How is the perfect square trinomial you obtained on the right side

related to x 5, which appears on the left side of the equation?

How do you know this?b) How could this be useful in finding an algebraic solution to the equation?

Tools

graphing calculator

grid paper

7.4 Techniques for Solving Logarithmic Equations MHR 387

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Example 1 Solve Simple Logarithmic Equations

Find the roots of each equation.

a) log (x 4) 1

b) log5 (2x 3) 2

Solution

a) Method 1: Use Algebraic Reasoning

log (x 4) 1

x 4 101 Rewrite in exponential form, using base 10.

x 10 4 Solve for x.

x 6

Method 2: Use Graphical Reasoning

Graph the left side and the right side as a linear-logarithmic system and

identify the x-coordinate of their point of intersection.

Let y1 log (x 4) and y

2 1.

Graph y1

by graphing y log x and

applying a horizontal translation of

4 units to the left.

Graph the horizontal line y2 1on the same grid, and identify the

x-coordinate of the point of intersection.

The two functions intersect when x 6.

Therefore, x 6 is the root of the

equation log (x 4) 1.

b) log5(2x 3) 2

log5(2x 3) log

525 Express the right side

as a base-5 logarithm.

2x 3 52

2x 25 3

2x 28

x 14

y

x2 4 624

2

4

2

4

0

y log x 4

y log x

y

x2 4 624

2

4

2

4

0

y log x 4y 1

C O N N E C T I O N S

If logm

a logm

b, then ab

for any base m, as shown below.

logm

a logm

b

logma_

logm

b 1

logba 1 Use the

change ofbase formula.

b1a Rewrite inexponential form.

ba

388 MHR Advanced Functions Chapter 7

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Example 2 Apply Factoring Strategies to Solve Equations

Solve. Identify and reject any extraneous roots.

a) log (x 1) 1 log (x 2)

b) log 3x2 48x 2_3Solution

a) log (x 1) 1 log (x 2)

log (x 1) log (x 2) 1 Isolate logarithmic terms on one side

of the equation.

log [(x 1)(x 2)] 1 Apply the product law of logarithms.

log (x2x 2) log 10 Expand the product of binomials on the

left side. Express the right side as a

common logarithm.

x2x 2 10

x2x 2 10 0

x2x 12 0 Express the quadratic equation in standard form.

(x 4)(x 3) 0 Solve the quadratic equation.

x4 or x 3

Looking back at the original equation, it is necessary to reject x4

as an extraneous root. Both log (x 1) and log (x 2) are undefined

for this value because the logarithm of a negative number is undefined.

Therefore, the only solution is x 3.

b) log3

x2 48x 2_3

log (x2 48x)1_

3 2_

3

1_3

log (x2 48x) 2_3 Write the radical as a power and apply the power law of logarithms.

log (x2 48x) 2 Multiply both sides by 3.

log (x2 48x) log 100 Express the right side as a common logarithm.

x2 48x 100

x2 48x 100 0

(x 50)(x 2) 0

x50 or x 2

Check these values for extraneous roots. For a valid solution, the argument

in green in the equation must be positive, and the left side must equal the

right side: log3

x2 48x 2_3

.


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Check x50:

3

x2 48x 3(50)2 48(50)

3

2500 2400

3

100L.S. log

3

x2 48x log

3

100 log 100

1_3

1_3

log 100

1_3

(2)

2_3

R.S.

3

100 0, and the solution satisfies the equation, so x50 is avalid solution.

Check x 2:

3

x2 48x 3(2)2 48(2)

3

100L.S. log

3

1001_

3log 100

1_3

(2)

2_3

R.S.

This is also a valid solution. Therefore, the roots of this equation are

x50 and x 2.

KEY CONCEPTS

It is possible to solve an equation involving logarithms by expressing

both sides as a logarithm of the same base: ifab, then log a log b,

and if log a log b, then ab.

When a quadratic equation is obtained, methods such as factoring or

applying the quadratic formula may be useful.

Some algebraic methods of solving logarithmic equations lead to

extraneous roots, which are not valid solutions to the original equation.


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Communicate Your Understanding

C1 Consider the equation log4(x 5) 2.

a) Which of the following expressions are equivalent to the right side

of the equation?

log2 4 log4 16 log5 25 log 100

b) Which one would you use to solve the equation for x, and why?

C2 Examine these graphing calculator screens.

a) What equation is being solved?b) What is the solution? Explain how you can tell.

C3 Consider the following statement: When solving a logarithmic equation

that results in a quadratic, you always obtain two roots: one valid and

one extraneous.

Do you agree or disagree with this statement? If you agree, explain why

it is correct. If you disagree, provide a counterexample.

A Practise

For help with questions 1 and 2, refer to Example 1.1. Find the roots of each equation. Check your

solutions using graphing technology.

a) log (x 2) 1

b) 2 log (x 25)

c) 4 2 log (p 62)

d) 1 log (w 7) 0

e) log (k 8) 2

f) 6 3 log 2n 0

2. Solve.

a) log3(x 4) 2

b) 5 log2(2x 10)

c) 2 log4(k 11) 0

d) 9 log5(x 100) 6

e) log8

(t 1) 1 0

f) log3

(n2 3n 5) 2

For help with questions 3 and 4, refer to Example 2.3. Solve. Identify and reject any extraneous roots.

Check your solutions using graphing technology.

a) log x log (x 4) 1

b) log x3 log 2 log (2x2)

c) log (v 1) 2 log (v 16)

d) 1 log y log (y 9)

e) log (k 2) log (k 1) 1

f) log (p 5) log (p 1) 3

4. Use Technology Refer to Example 2b). Verify

the solutions to the equation using graphing

technology. Explain your method.


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C Extend and Challenge

12. Solve the equation 2 log (x 11) (1_2)x

.

Explain your method.

13. Show that iflog

bac and

logybc, then

logayc2.

14. Math Contest Find the minimum value

of 1 2 3 4 5 6 7 8 9,

where represents either or .

15. Math Contest Given that 3 m 1_m ,

determine the value ofm 1_m.16. Math Contest Let u and v be two positive

real numbers satisfying the two equations

uvuv 10 and u2v2 40. What is

the value of the integer closest to uv?

A 4 B 5 C 6 D 7 E 8

Connecting

Problem Solving

Reasoning and Proving

Refecting

Selecting ToolsRepresenting

Communicating

B Connect and Apply

5. Solve. Check for extraneous roots. Check your

results using graphing technology.

a) log x2 3x 1_2

b) log x2 48x 16. Solve. Identify any extraneous roots.

a) log2(x 5) log

2(2x) 8

b) log (2k 4) 1 log k

7. Use Technology Find the roots of each equation,

correct to two decimal places, using graphing

technology. Sketch the graphical solution.

a) log (x 2) 2 log x

b) 3 log (x 2) log (2x) 3

8. Chapter Problem At a concert, the loudness of

sound, L, in decibels, is given by the equation

L 10 log I_

I

0

, where Iis the intensity, in

watts per square metre, and I0

is the minimum

intensity of sound audible to the average

person, or 1.0 1012 W/m2.

a) The sound intensity at the concert is

measured to be 0.9 W/m2. How loud is

the concert?

b) At the concert, the person beside you

whispers with a loudness of 20 dB. What

is the whispers intensity?

c) On the way home from the concert, your

car stereo produces 120 dB of sound.

What is its intensity?

9. a) Is the following statement true?

log (3) log (4) log 12

Explain why or why not.

b) Is the following statement true?

log 3 log 4 log 12

Explain why or why not.

10. The aperture setting, or f-stop, of a digital

camera controls the amount of light exposure

on the sensor. Each higher number of the

f-stop doubles the amount of light exposure.

The formula n log21_p represents the change

in the number, n, of the f-stop needed, wherep

is the amount of light exposed on the sensor.

a) A photographer wishes to change the f-stop

to accommodate a cloudy day in which

only 1_4

of the sunlight is available. How many

f-stops does the setting need to be moved?

b) If the photographer decreases the f-stop

by four settings, what fraction of light is

allowed to fall on the sensor?

11. a) Solve and check for

any extraneous roots.

2_3

log3

w2 10wb) Solve the equation in

part a) graphically.

Verify that the graphical and algebraic

solutions agree.

C O N N E C T I O N S

You used the decibel scale in Chapter 6. Refer to Section 6.5.

Connecting

Problem Solving

Reasoning and Proving

Refecting

Selecting ToolsRepresenting

Communicating


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Section 7.4 Page 387 to 392