Download Solving Exponential and Logarithmic Equations 5.5 and more Summaries Mathematics in PDF only on Docsity!
Section 5.5 Solving Exponential and Logarithmic Equations 281
Solving Exponential and Logarithmic Equations
Work with a partner. Match each equation with the graph of its related system of equations. Explain your reasoning. Then use the graph to solve the equation. a. e x^ = 2 b. ln x = − 1 c. 2 x^ = 3 − x^ d. log 4 x = 1 e. log 5 x = 1 — 2 f. 4 x^ = 2
Solving Exponential and Logarithmic Equations
Work with a partner. Look back at the equations in Explorations 1(a) and 1(b). Suppose you want a more accurate way to solve the equations than using a graphical approach. a. Show how you could use a numerical approach by creating a table. For instance, you might use a spreadsheet to solve the equations. b. Show how you could use an analytical approach. For instance, you might try solving the equations by using the inverse properties of exponents and logarithms.
Communicate Your AnswerCommunicate Your Answer
3. How can you solve exponential and logarithmic equations? 4. Solve each equation using any method. Explain your choice of method. a. 16 x^ = 2 b. 2 x^ = 42 x^ +^1 c. 2 x^ = 3 x^ +^1 d. log x = 1 — 2
e. ln x = 2 f. log 3 x = 3 — 2
MAKING SENSE
OF PROBLEMS
To be proficient in math, you need to plan a solution pathway rather than simply jumping into a solution attempt.
Essential QuestionEssential Question How can you solve exponential and
logarithmic equations?
Solving Exponential and
Logarithmic Equations
A.
x
y 4
− 4
− 2
− 4 − 2 2 4
B.
x
y 4
2
− 4
− 2
− 4 − 2 2 4
C.
x
y 4
− 4
− 2
− 4 − 2 2 4
D.
x
y 4
− 4
− 2
− 4 − 2 2 4
E.
x
y 4
2
− 4
− 2
− 4 − 2 2 4
F.
x
y 4
2
− 2
− 2 2
282 Chapter 5 Exponential and Logarithmic Functions
5.5 Lesson^ What You Will LearnWhat You Will Learn
Solve exponential equations. Solve logarithmic equations. Solve exponential and logarithmic inequalities.
Solving Exponential Equations Exponential equations are equations in which variable expressions occur as exponents. The result below is useful for solving certain exponential equations.
The preceding property is useful for solving an exponential equation when each side of the equation uses the same base (or can be rewritten to use the same base). When it is not convenient to write each side of an exponential equation using the same base, you can try to solve the equation by taking a logarithm of each side.
Solving Exponential Equations
Solve each equation.
a. 100 x^ = (^) (
— 10 )
x − 3 b. 2 x^ = 7
SOLUTION
a. 100 x^ = (^) (
— 10 )
x − 3 Write original equation.
( 10 2 ) x^ = ( 10 −^1 ) x^ −^3 Rewrite 100 and
— 10 as powers with base 10. 102 x^ = 10 − x^ +^3 Power of a Power Property 2 x = − x + 3 Property of Equality for Exponential Equations x = 1 Solve for x.
b. 2 x^ = 7 Write original equation. log 2 2 x^ = log 2 7 Take log 2 of each side. x = log 2 7 log b b x^ = x x ≈ 2.807 Use a calculator.
Check
Enter y = 2 x^ and y = 7 in a graphing calculator. Use the intersect feature to find the intersection point of the graphs. The graphs intersect at about (2.807, 7). So, the solution of 2 x^ = 7 is about 2.807. ✓
Check
1001 =
(
— 10 )
1 − 3
(
— 10 )
− 2
100 = 100 ✓
exponential equations, p. 282 logarithmic equations, p. 283 Previous extraneous solution inequality
Core VocabularyCore Vocabullarry
CoreCore ConceptConcept
Property of Equality for Exponential Equations
Algebra If b is a positive real number other than 1, then b x^ = b y^ if and only if x = y. Example If 3 x^ = 35 , then x = 5. If x = 5, then 3 x^ = 35.
5
− 3
0
10
Intersection X=2.8073549 Y=
284 Chapter 5 Exponential and Logarithmic Functions
Solving Logarithmic Equations
Solve (a) ln(4 x − 7) = ln( x + 5) and (b) log 2 (5 x − 17) = 3.
SOLUTION
a. ln(4 x − 7) = ln( x + 5) Write original equation. 4 x − 7 = x + 5 Property of Equality for Logarithmic Equations 3 x − 7 = 5 Subtract x from each side. 3 x = 12 Add 7 to each side. x = 4 Divide each side by 3.
b. log2(5 x − 17) = 3 Write original equation. 2 log^2 (5 x^ −^ 17)^ = 23 Exponentiate each side using base 2. 5 x − 17 = 8 b log b^ x^ = x 5 x = 25 Add 17 to each side. x = 5 Divide each side by 5.
Because the domain of a logarithmic function generally does not include all real numbers, be sure to check for extraneous solutions of logarithmic equations. You can do this algebraically or graphically.
Solving a Logarithmic Equation
Solve log 2 x + log( x − 5) = 2.
SOLUTION
log 2 x + log( x − 5) = 2 Write original equation. log[2 x ( x − 5)] = 2 Product Property of Logarithms 10 log[2 x ( x^ −^ 5)]^ = 102 Exponentiate each side using base 10. 2 x ( x − 5) = 100 b log b^ x^ = x 2 x^2 − 10 x = 100 Distributive Property 2 x^2 − 10 x − 100 = 0 Write in standard form. x^2 − 5 x − 50 = 0 Divide each side by 2. ( x − 10)( x + 5) = 0 Factor. x = 10 or x = − 5 Zero-Product Property
The apparent solution x = −5 is extraneous. So, the only solution is x = 10.
Monitoring Progress (^) Help in English and Spanish at BigIdeasMath.com
Solve the equation. Check for extraneous solutions.
5. ln(7 x − 4) = ln(2 x + 11) 6. log 2 ( x − 6) = 5 7. log 5 x + log( x − 1) = 2 8. log 4 ( x + 12) + log 4 x = 3
Check
ln(4 (^) ⋅ 4 − 7) =
ln(4 + 5)
ln(16 − 7) =
ln 9
ln 9 = ln 9 ✓
Check
log 2 (5 (^) ⋅ 5 − 17) =
log 2 (25 − 17) =
log 2 8 =
Because 2^3 = 8, log 2 8 = 3. ✓
Check
log(2 (^) ⋅ 10) + log(10 − 5) =
log 20 + log 5 =
log 100 =
2 = 2 ✓
log[2 (^) ⋅ (−5)] + log(− 5 − 5) =
log(−10) + log(−10) =
Because log(−10) is not defi ned,
−5 is not a solution. ✗
Section 5.5 Solving Exponential and Logarithmic Equations 285
Solving Exponential and Logarithmic Inequalities Exponential inequalities are inequalities in which variable expressions occur as exponents, and logarithmic inequalities are inequalities that involve logarithms of variable expressions. To solve exponential and logarithmic inequalities algebraically, use these properties. Note that the properties are true for ≤ and ≥. Exponential Property of Inequality: If b is a positive real number greater than 1, then bx^ > by^ if and only if x > y , and bx^ < by^ if and only if x < y. Logarithmic Property of Inequality: If b , x , and y are positive real numbers with b > 1, then log b x > log b y if and only if x > y , and log b x < log b y if and only if x < y. You can also solve an inequality by taking a logarithm of each side or by exponentiating.
Solving an Exponential Inequality
Solve 3 x^ < 20.
SOLUTION
3 x^ < 20 Write original inequality. log 3 3 x^ < log 3 20 Take log 3 of each side. x < log 3 20 log b bx^ = x
The solution is x < log 3 20. Because log 3 20 ≈ 2.727, the approximate solution is x < 2.727.
Solving a Logarithmic Inequality
Solve log x ≤ 2.
SOLUTION
Method 1 Use an algebraic approach. log x ≤ 2 Write original inequality. 10 log^10 x^ ≤ 102 Exponentiate each side using base 10. x ≤ 100 b log b x^ = x
Because log x is only defined when x > 0, the solution is 0 < x ≤ 100.
Method 2 Use a graphical approach. Graph y = log x and y = 2 in the same viewing window. Use the intersect feature to determine that the graphs intersect when x = 100. The graph of y = log x is on or below the graph of y = 2 when 0 < x ≤ 100.
The solution is 0 < x ≤ 100.
Monitoring ProgressMonitoring Progress (^) Help in English and Spanish at BigIdeasMath.com
Solve the inequality.
9. ex^ < 2 10. 102 x^ −^6 > 3 11. log x + 9 < 45 12. 2 ln x − 1 > 4
STUDY TIP
Be sure you understand that these properties of inequality are only true for values of b > 1.
175
− 1
− 50
3
Intersection X=100 Y=
Section 5.5 Solving Exponential and Logarithmic Equations 287
In Exercises 33–40, solve the equation. Check for extraneous solutions. (See Example 4.)
33. log2 x + log 2 ( x − 2) = 3 34. log6 3 x + log 6 ( x − 1) = 3 35. ln x + ln( x + 3) = 4 36. ln x + ln( x − 2) = 5 37. log3 3 x^2 + log3 3 = 2 38. log4(− x ) + log 4 ( x + 10) = 2 39. log3( x − 9) + log 3 ( x − 3) = 2 40. log5( x + 4) + log 5 ( x + 1) = 2
ERROR ANALYSIS In Exercises 41 and 42, describe and correct the error in solving the equation.
41. log 3 (5x − 1) = 4 3 log^3 (5x^ −^ 1)^ = 43 5 x − 1 = 64 5 x = 65 x = 13
log 4 (x + 12) + log 4 x = 3 log 4 [(x + 12)(x)] = 3 4 log^4 [(x^ +^ 12)(x)]^ = 43 (x + 12)(x) = 64 x^2 + 12 x − 64 = 0 (x + 16)(x − 4) = 0 x = − 16 or x = 4
43. PROBLEM SOLVING You deposit $100 in an account that pays 6% annual interest. How long will it take for the balance to reach $1000 for each frequency of compounding? a. annual b. quarterly c. daily d. continuously 44. MODELING WITH MATHEMATICS The apparent magnitude of a star is a measure of the brightness of the star as it appears to observers on Earth. The apparent magnitude M of the dimmest star that can be seen with a telescope is M = 5 log D + 2, where D is the diameter (in millimeters) of the telescope’s objective lens. What is the diameter of the objective lens of a telescope that can reveal stars with a magnitude of 12? 45. ANALYZING RELATIONSHIPS Approximate the solution of each equation using the graph. a. 1 − 55 −^ x^ = − 9 b. log 2 5 x = 2
x
y 2
− 12
− 4
8
y = 1 − 55 −^ x
y = − 9
x
y
4
8
12
− 4
2 4 y = log 2 5 x
y = 2
46. MAKING AN ARGUMENT Your friend states that a logarithmic equation cannot have a negative solution because logarithmic functions are not defined for negative numbers. Is your friend correct? Justify your answer.
In Exercises 47–54, solve the inequality. (See Examples 5 and 6.)
47. 9 x^ > 54 48. 4 x^ ≤ 36 49. ln x ≥ 3 50. log 4 x < 4 51. 34 x^ −^5 < 8 52. e^3 x^ +^4 > 11 53. −3 log 5 x + 6 ≤ 9 54. −4 log 5 x − 5 ≥ 3 55. COMPARING METHODS Solve log 5 x < 2 algebraically and graphically. Which method do you prefer? Explain your reasoning. 56. PROBLEM SOLVING You deposit $1000 in an account that pays 3.5% annual interest compounded monthly. When is your balance at least $1200? $3500? 57. PROBLEM SOLVING An investment that earns a rate of return r doubles in value in t years, where t = —ln 2 ln(1 + r )
and r is expressed as a decimal. What rates of return will double the value of an investment in less than 10 years?
58. PROBLEM SOLVING Your family purchases a new car for $20,000. Its value decreases by 15% each year. During what interval does the car’s value exceed $10,000?
USING TOOLS In Exercises 59–62, use a graphing calculator to solve the equation.
59. ln 2 x = 3 − x^ +^2 60. log x = 7 − x 61. log x = 3 x^ −^3 62. ln 2 x = e x^ −^3
288 Chapter 5 Exponential and Logarithmic Functions
36 cm
32 cm
28 cm
24 cm
63. REWRITING A FORMULA A biologist can estimate the age of an African elephant by measuring the length of its footprint and using the equation = 45 − 25.7 e −0.09 a , where is the length (in centimeters) of the footprint and a is the age (in years). a. Rewrite the equation, solving for a in terms of. b. Use the equation in part (a) to fi nd the ages of the elephants whose footprints are shown. 64. HOW DO YOU SEE IT? Use the graph to approximate the solution of the inequality 4 ln x + 6 > 9. Explain your reasoning.
x
y 12
6
− 2 2 4 6
y = (^9) y = 4 ln x + 6
65. OPEN-ENDED Write an exponential equation that has a solution of x = 4. Then write a logarithmic equation that has a solution of x = −3. 66. THOUGHT PROVOKING Give examples of logarithmic or exponential equations that have one solution, two solutions, and no solutions.
CRITICAL THINKING In Exercises 67–72, solve the equation.
67. 2 x^ +^3 = 53 x^ −^1 68. 103 x^ −^8 = 25 −^ x 69. log3( x − 6) = log 9 2 x 70. log4 x = log 8 4 x 71. 22 x^ − (^12) ⋅ 2 x^ + 32 = 0 72. 52 x^ + (^20) ⋅ 5 x^ − 125 = 0 73. WRITING In Exercises 67–70, you solved exponential and logarithmic equations with different bases. Describe general methods for solving such equations. 74. PROBLEM SOLVING When X-rays of a fixed wavelength strike a material x centimeters thick, the intensity I ( x ) of the X-rays transmitted through the material is given by I ( x ) = I 0 e −μ x , where I 0 is the initial intensity and μ is a value that depends on the type of material and the wavelength of the X-rays. The table shows the values of μ for various materials and X-rays of medium wavelength.
Material Aluminum Copper Lead
Value of μ 0.43 3.2 43
a. Find the thickness of aluminum shielding that reduces the intensity of X-rays to 30% of their initial intensity. ( Hint : Find the value of x for which I ( x ) = 0.3 I 0 .) b. Repeat part (a) for the copper shielding. c. Repeat part (a) for the lead shielding. d. Your dentist puts a lead apron on you before taking X-rays of your teeth to protect you from harmful radiation. Based on your results from parts (a)–(c), explain why lead is a better material to use than aluminum or copper.
Maintaining Mathematical ProficiencyMaintaining Mathematical Proficiency
Write an equation in point-slope form of the line that passes through the given point and has the given slope. (Skills Review Handbook)
75. (1, −2); m = 4 76. (3, 2); m = − 2 77. (3, −8); m = − —^13 78. (2, 5); m = 2
Use fi nite differences to determine the degree of the polynomial function that fits the data. Then use technology to find the polynomial function. (Section 3.9)
79. (−3, −50), (−2, −13), (−1, 0), (0, 1), (1, 2), (2, 15), (3, 52), (4, 125) 80. (−3, 139), (−2, 32), (−1, 1), (0, −2), (1, −1), (2, 4), (3, 37), (4, 146) 81. (−3, −327), (−2, −84), (−1, −17), (0, −6), (1, −3), (2, −32), (3, −189), (4, −642)
Reviewing what you learned in previous grades and lessons
