Download 7-9 Exponential and Logarithmic Equations

7-9
Exponential and Logarithmic Equations
TEKS FOCUS
VOCABULARY
TEKS (5)(D) Solve exponential equations
of the form y = abx where a is a nonzero
real number and b is greater than zero and
not equal to one and single logarithmic
equations having real solutions.
ĚExponential equation – An exponential equation contains the form
TEKS (1)(C) Select tools, including real
objects, manipulatives, paper and pencil, and
technology as appropriate, and techniques,
including mental math, estimation, and
number sense as appropriate, to solve
problems.
ĚNumber sense – the understanding of what numbers mean and how
bcx , with the exponent including a variable.
ĚLogarithmic equation – A logarithmic equation is an equation that
includes a logarithm involving a variable.
they are related
Additional TEKS (1)(A), (5)(B), (5)(C),
(5)(E)
ESSENTIAL UNDERSTANDING
You can use logarithms to solve exponential equations. You can use exponents to
solve logarithmic equations.
Problem 1
P
Solving an Exponential Equation—Common Base
Multiple Choice What is the solution of 163x = 8?
What common base is
appropriate?
2 because 16 and 8 are
both powers of 2.
3
x = 14
x=7
(24)3x = 23
=
23
12x = 3
x = 14
Rewrite the terms with a common base.
Power Property of Exponents
If two numbers with the same base are equal, their exponents are equal.
Solve and simplify.
The correct answer is A.
Lesson 7-9
x=4
163x = 8
212x
318
x=1
Exponential and Logarithmic Equations
Problem 2
P
Solving an Exponential Equation—Different Bases
What is the solution of 153x = 285?
Which property of
logarithms will help
isolate x?
The rule log ax = x log a
moves x out of the
exponent position.
153x = 285
log 153x = log 285
3x log 15 = log 285
Take the logarithm of each side.
Power Property of Logarithms
log 285
x = 3 log 15
Divide each side by 3 log 15 to isolate x.
x ≈ 0.6958
Use a calculator.
Check 153x = 285
153(0.6958) ≈ 285.0840331 ≈ 285 ✓
Problem
bl
3
TEKS Process Standard (1)(C)
Solving an Exponential Equation With a Graph or Table
What is the solution of 43x = 6000?
Method 1 Solve using a graph.
Use a graphing calculator. Graph the
equations.
Y1 = 43x
Y2 = 6000
WINDOW
Xmin = –5
Xmax = 6
Xscl = .1
Ymin = –500
Ymax = 7000
Yscl = 400
Xres = 1
Intersection
X⫽2.0917911
Y⫽6000
Adjust the window to find the point
of intersection. The solution is x ≈ 2.09.
Method 2 Solve using a table.
How do you choose
TblStart and ΔTbl
values?
Start with 0 and 1,
respectively. Adjust both
values as you close in on
the solution.
Use the table feature of a graphing
calculator.
Enter Y1 = 43x .
c
Use
U the TABLE SETUP and 𝚫Tbl
features
to locate the x-value that gives
f
the
t y-value closest to 6000.
TABLE SETUP
TblStart = 2.05
Tbl = .01
Indpnt: Auto Ask
Depend: Auto Ask
X
2.05
2.06
2.07
2.08
2.09
2.1
2.11
Y1
5042.8
5256.9
5480.2
5712.9
5955.5
6208.4
6472
Y1=5955.47143094
The
T solution is x ≈ 2.09.
PearsonTEXAS.com
319
Problem 4
P
TEKS Process Standard (1)(A)
Modeling With an Exponential Equation
STEM
Resource Management Wood is a sustainable, renewable, natural resource when
you manage forests properly. Your lumber company has 1,200,000 trees. You plan
to harvest 7% of the trees each year. How many years will it take to harvest half of
the trees?
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Number of years
it takes to harvest
600,000 trees
Step 1 Is an exponential model reasonable for this situation?
What equation
should you use to
model this situation?
Since you are planning to
harvest 7% of the trees
each year, you should use
y = abx, where b is the
decay factor.
Yes, you are harvesting a fixed percentage each year.
Step 2 Define the variables and determine the model.
S
Let n = the number of years it takes to harvest half of the trees.
Let T (n) = the number of trees remaining after n years.
A reasonable model is T (n) = a(b)n.
Step 3 Use the model to write an exponential equation.
S
T (n) = 600,000
a = 1,200,000
r = -7% = -0.07
b = 1 + r = 1 + ( -0.07) = 0.93
So, 1,200,000(0.93)n = 600,000.
Step 4 Solve the equation. Use logarithms.
1,200,000(0.93)n = 600,000
600,000
0.93n = 1,200,000
log 0.93n = log 0.5
n log 0.93 = log 0.5
Isolate the term with n.
Take the logarithm of each side.
Power Property of Logarithms
log 0.5
n = log 0.93
Solve for n.
n ≈ 9.55
Use a calculator.
It will take about 9.55 years to harvest half of the original trees.
320
Lesson 7-9
Exponential and Logarithmic Equations
Problem 5
P
Solving a Logarithmic Equation
What is the solution of log (4x − 3) = 2?
Method
1 Solve using exponents.
M
How do you convert
between log form
and exponential
form?
Use the rule: log a = b
if and only if a = 10b.
log (4x - 3) = 2
4x - 3 = 102
:ULWHLQH[SRQHQWLDOIRUP
4x = 103
Simplify.
103
x = 4 = 25.75
Method 2 Solve using a graph.
Graph the equations
Y1 = LOG (4x − 3) and Y2 = 2.
Find the point of intersection.
The solution is x = 25.75.
Solve for x.
Method 3 Solve using a table.
Enter Y1 = LOG (4x − 3).
Use the TABLE SETUP feature to find the
x-value that corresponds to a y-value of
2 in the table.
The solution is x = 25.75.
TABLE SETUP
TblStart = 25
Tbl = .01
Indpnt: Auto Ask
Depend: Auto Ask
Intersection
X⫽25.75
Y⫽2
X
25.7
25.71
25.72
25.73
25.74
25.75
25.76
Y1=2
Y1
1.9991
1.9993
1.9995
1.9997
1.9998
2
2.0002
Problem
bl
6
Using Logarithmic Properties to Solve an Equation
What is the solution of log (x − 3) + log x = 1?
log (x - 3) + log x = 1
log ((x - 3)x) = 1
(x - 3)x =
x2
101
- 3x - 10 = 0
(x - 5)(x + 2) = 0
x=5
or
x = -2
Product Property of Logarithms
:ULWHLQH[SRQHQWLDOIRUP
Simplify to a quadratic equation in standard form.
Factor the trinomial.
Solve for x.
Check. Determine the reasonableness of the solutions.
What is the domain
of the logarithmic
function?
Logs are defined only for
positive numbers. The log
of a negative number is
undefined.
log (x - 3) + log (x) = 1
log (-2 - 3) + log (-2) ≟ 1 ✘
log (x - 3) + log (x) = 1
log (5 - 3) + log (5) ≟ 1
log 2 + log 5 ≟ 1
0.3010 + 0.6990 = 1 ✔
IIf log (x - 3) + log(x) = 1, x = 5. The solution x = -2 is not reasonable because the
llog of a negative number is undefined.
PearsonTEXAS.com
321
Problem 7
P
TEKS Process Standard (1)(A)
Modeling With Logarithms
The enrollment at a private school was initially 120 students, and has been increasing
by about 30% each year. Formulate a logarithmic function t(x) that approximates the
number of years t it will take before the enrollment reaches x students.
Step 1 Write an exponential model.
The initial enrollment is 120, and the growth rate is 30%. An exponential
model for this situation is x(t) = 120(1.3)t .
Step 2 Solve the exponential equation for t.
x = 120(1.3)t
log x =
Exponential model
log [120(1.3)t]
Take the logarithm of each side.
log x = log 120 + t log 1.3
t log 1.3 = log x - log 120
Properties of logarithms
Exponential model
log x - log 120
t=
log 1.3
Solve for t.
Step 3 Write the function.
log x - log 120
NLINE
HO
ME
RK
O
The function t(x) =
approximates the number of years t
log 1.3
required to reach an enrollment of x students.
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Solve each equation.
1. 2x = 8
For additional support when
completing your homework,
go to PearsonTEXAS.com.
2. 32x = 27
1
4. 53x = 125
3. 43x = 64
Rewrite each exponential function as a logarithmic function.
6. A(t) = 500(1.04)0.5t +1
5. P(t) = 100(0.85)t
7. T = T0e -1.6t
8. The magnitude of a star measures its brightness. Each decrease of 1 in
magnitude represents a star that appears b times brighter. The ratio of first
magnitude to sixth magnitude is 100 to 1. Write a logarithmic function and use
it to find b to the nearest thousandth. Show your work.
Solve each equation. Round to the nearest ten-thousandth. Check your answers.
9. 2x = 3
10. 4x = 19
11. 252x+1 = 144
12. 5 - 3x = -40
x
13. Consider the equation 23 = 80.
a. Solve the equation by taking the logarithm base 10 of each side.
b. Solve the equation by taking the logarithm base 2 of each side.
c. Explain Mathematical Ideas (1)(G) Compare your result in parts (a) and (b).
What are the advantages of each method? Explain.
322
Lesson 7-9
Exponential and Logarithmic Equations
14. Apply Mathematics (1)(A) An earthquake of magnitude 8.7 hit
Sumatra in March 2005. In December 2004 an earthquake about
4 times as strong as the Sumatra 2005 earthquake hit off the
northwest coast of Sumatra. Find the magnitude of the 2004
earthquake.
Magnitude 8.7
The Richter Scale
magnitude: ⴙ 1
0
E
1
2
3
4
5
6
7
8
9
E∙30
E∙302
E∙303
E∙304
E∙305
E∙306
E∙307
E∙308
E∙309
Epicenter,
March 2005
Sumatra
S
Sumat
umatra
ra
energy released: ⴛ 30
15. As a town gets smaller, the population of its high school decreases by 6% each
eacch
year. The senior class has 160 students now. In how many years will it have
ve
about 100 students? Write an equation. Then solve the equation without
graphing.
Solve by graphing. Round to the nearest ten-thousandth.
16. 47x = 250
17. 53x = 500
18. 6x = 4565
19. 1.5x = 356
Use a table to solve each equation. Round to the nearest hundredth.
20. 2x+3 = 512
21. 3x-1 = 72
22. 62x = 10
23. 52x = 56
24. The equation y = 6.72(1.014)x models the world population y, in billions of
people, x years after the year 2000. Find the year in which the world population
is about 8 billion.
25. Apply Mathematics (1)(A) The table below lists the states with the highest and
with the lowest population growth rates. Determine in how many years each
event can occur. Use the model P = P0(1 + r)x, where P0 is population from the
table, as of July 2007; x is the number of years after July 2007, P is the projected
population, and r is the growth rate.
a. Population of Idaho exceeds 2 million.
b. Population of Michigan decreases by 1 million.
c. Population of Nevada doubles.
State
Growth
rate (%)
Population
(in thousands)
Growth
rate (%)
Population
(in thousands)
1. Nevada
2.93
2,565
46. New York
0.08
19,298
2. Arizona
2.81
6,339
47. Vermont
0.08
621
3. Utah
4. Idaho
2.55
2,645
48. Ohio
0.03
11,467
2.43
1,499
49. Michigan
⫺0.30
10,072
5. Georgia
2.17
9,545
50. Rhode Island
⫺0.36
1,058
State
SOURCE: U.S. Census Bureau
PearsonTEXAS.com
323
Select Tools to Solve Problems (1)(C) Solve each equation.
Check your answers.
26. log 2x = -1
27. log (3x + 1) = 2 28. log x + 4 = 8
29. 3 log x = 1.5
Solve each equation. Determine the reasonableness of the solutions.
30. log x - log 3 = 8
STEM
31. log 2x + log x = 11
32. log (7x + 1) = log (x - 2) + 1
33. Apply Mathematics (1)(A) The pitch,
tch, or
frequency, of a piano note is
related to its position on the
keyboard by the
function
n
F (n) = 440 212, where F is the
frequency of the sound waves in
cycles per second and n is the
number of piano keys above or
below Concert A, as shown. If
n = 0 at Concert A, which of the
instruments shown in the
diagram can sound notes at the
given frequency?
Bassoon
Guitar
Harp
Violin
Viola
#
a. 590
STEM
b. 120
Cello
Bass
–10
c. 1440
–5
0
5
10
d. 2093
Apply Mathematics (1)(A) In Exercise 34, the loudness measured in decibels (dB)
is defined by loudness = 10 log II , where I is the intensity and I0 = 10−12 W>m2.
0
34. The human threshold for pain is 120 dB. Instant perforation of the eardrum
occurs at 160 dB.
a. Find the intensity of each sound.
b. How many times as intense is the noise that will perforate an eardrum as the
noise that causes pain?
STEM
()
h
35. Apply Mathematics (1)(A) In the formula P = P0 12 4795, P is the atmospheric
pressure in millimeters of mercury at elevation h meters above sea level. P0 is
the atmospheric pressure at sea level. If P0 equals 760 mm, at what elevation is
the pressure 42 mm?
TEXAS Test Practice
T
36. The graph at the right shows the translation of the graph of the
parent function y = 0 x 0 down 2 units and 3 units to the right.
What is the area of the shaded triangle in square units?
37. What does x equal if log (1 + 3x) = 3?
38. Using the change of base formula, what is the value of x for which
log 9 x = log 3 5?
324
Lesson 7-9
Exponential and Logarithmic Equations
y
x