Section 7.4 Page 387 to 392
Transcript of Section 7.4 Page 387 to 392
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8/9/2019 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
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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
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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
392 MHR Advanced Functions Chapter 7