49. Limiting Magnitude of a Telescope A telescope is limited in its usefulness by the brightness of the star that it is aimed at and by the diameter of its lens. One measure of a star's brightness is its magnitude; the dimmer the star, the larger its magnitude. A formula for the limiting magnitude L of a telescope-that is, the magnitude of the dimmest star that it can be used to view–is given by L = 9 + 5.1 log d where d is the diameter (in inches) of the lens (a) What is the limiting magnitude of a 3.5-inch telescope? (b) What diameter is required to view a star of magnitude 14
49. Limiting Magnitude of a Telescope A telescope is limited in its usefulness by the brightness of the star that it is aimed at and by the diameter of its lens. One measure of a star's brightness is its magnitude; the dimmer the star, the larger its magnitude. A formula for the limiting magnitude L of a telescope-that is, the magnitude of the dimmest star that it can be used to view–is given by L = 9 + 5.1 log d where d is the diameter (in inches) of the lens (a) What is the limiting magnitude of a 3.5-inch telescope? (b) What diameter is required to view a star of magnitude 14
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Concept explainers
Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
Topic Video
Question
49
![In Problems l1-14, each function is one-to-one. Find the inverse of each function and check your answer. Find the domain and range of f and f1
(a) What is the limiting magnitude of a 3.5-inch telescope?
356 CHAPTER 5 Exponential and Logarithmic Functions
14. f(x) = x + 1
2r + 3
13. f(x) = Vx – 2
11. f(x) =
12. f(x) =
5x - 2
15. Given f(x) = 3" and g(x)= loga x, evaluate each of the following.
(d)
(a) f(4)
(b) g(9)
(c) f(-2)
16. Change 52 = z to an equivalent statement involving a logarithm.
17. Change logs u = 13 to an equivalent statement involving an exponent.
In Problems 18 and 19, find the domain of each logarithmic function.
19. H(x) = log2 (x² – 3x + 2)
18. f(x) = log (3x – 2)
In Problems 20-22, find the exact value of each expression, Do not use a calculator.
22. 2lo820.4
20. log2
21. In eV
In Problems 23-26, write each expression as the sum and/or difference of logarithms. Express powers as factors.
24. log2 (a? Vb)“ a > 0, b > 0
23. log,()
uv
u > 0, v > 0, w >0
2x + 3
26. In 2- 3x + 2)
x > 2
25. log (x? Vx + 1) x > 0
In Problems 27-29, write each expression as a single logarithm.
log, Va
In) + In) - In(x² – 1)
27. 3 log4 x? +
국n (12 + 1)-4In -In(x-4) + Inx]
29.
30. Use the Change-of-Base Formula and a calculator to evaluate log, 19. Round your answer to three decimal places.
31. Graph y = log3 x using a graphing utility and the Change-of-Base Formula.
In Problems 32–35, for each function f:
(a) Find the domain of f.
(d) Find f-1, the inverse function of f.
(c) From the graph, determine the range and any asymptotes of f.
() Graph f1.
(b) Graph f.
(e) Find the domain and the range of f.
32. f(x) = 2"-3
33. f(x) = 1 + 3-*
34. f(x) = 3e*-2
35. f(x) = ;
x + 3)
In Problems 36-46, solve each equation. Express irrational solutions in exact form.
36. 86+3x = 4
37. 3*+* = V3
38. log, 64 = -3
39, 5* = 3*+2
40. 252x = 5-12
41. log3 Vx – 2 = 2
42. 8 = 4. 25x
43. 2 .5 = 10"
44. log, (x + 3) + log, (x + 4) = 1
45. el-x = 5
46. 9* + 4.3* - 3 = 0
47. Suppose that f(x) = log2(x – 2) + 1.
(a) Graph f.
(b) What is f(6)? What point is on the graph of f ?
(c) Solve f(x) = 4. What point is on the graph of f?
(d) Based on the graph drawn in part (a), solve f(x) > 0.
(e) Find f (x). Graph f on the same Cartesian plane
49. Limiting Magnitude of a Telescope A telescope is limile
in its usefulness by the brightness of the star that it is ail
at and by the diameter of its lens One measure of a stai s
brightness is its magnitude: the dimmer the star, the lalge
its magnitude. A formula for the limiting magnitude L
telescope-that is, the magnitude of the dimmest star tie
can be used to view-is given by
as f.
48. Amplifying Sound An amplifier's power output P (in watts)
is related to its decibel voltage gain d by the formula
L = 9 + 5.1 log d
where d is the diameter (in inches) of the lens.
P = 25e0.ld
(a) Find the power output for a decibel voltage gain
of 4 decibels.
(b) For a power output of 50 watts, what is the decibel
voltage gain?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3fcc2212-54a6-4f1d-b6b8-1f7f4adab2ee%2Fd849e10d-3f66-4839-8d8c-166ee9dc7574%2Ft1dzrpk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:In Problems l1-14, each function is one-to-one. Find the inverse of each function and check your answer. Find the domain and range of f and f1
(a) What is the limiting magnitude of a 3.5-inch telescope?
356 CHAPTER 5 Exponential and Logarithmic Functions
14. f(x) = x + 1
2r + 3
13. f(x) = Vx – 2
11. f(x) =
12. f(x) =
5x - 2
15. Given f(x) = 3" and g(x)= loga x, evaluate each of the following.
(d)
(a) f(4)
(b) g(9)
(c) f(-2)
16. Change 52 = z to an equivalent statement involving a logarithm.
17. Change logs u = 13 to an equivalent statement involving an exponent.
In Problems 18 and 19, find the domain of each logarithmic function.
19. H(x) = log2 (x² – 3x + 2)
18. f(x) = log (3x – 2)
In Problems 20-22, find the exact value of each expression, Do not use a calculator.
22. 2lo820.4
20. log2
21. In eV
In Problems 23-26, write each expression as the sum and/or difference of logarithms. Express powers as factors.
24. log2 (a? Vb)“ a > 0, b > 0
23. log,()
uv
u > 0, v > 0, w >0
2x + 3
26. In 2- 3x + 2)
x > 2
25. log (x? Vx + 1) x > 0
In Problems 27-29, write each expression as a single logarithm.
log, Va
In) + In) - In(x² – 1)
27. 3 log4 x? +
국n (12 + 1)-4In -In(x-4) + Inx]
29.
30. Use the Change-of-Base Formula and a calculator to evaluate log, 19. Round your answer to three decimal places.
31. Graph y = log3 x using a graphing utility and the Change-of-Base Formula.
In Problems 32–35, for each function f:
(a) Find the domain of f.
(d) Find f-1, the inverse function of f.
(c) From the graph, determine the range and any asymptotes of f.
() Graph f1.
(b) Graph f.
(e) Find the domain and the range of f.
32. f(x) = 2"-3
33. f(x) = 1 + 3-*
34. f(x) = 3e*-2
35. f(x) = ;
x + 3)
In Problems 36-46, solve each equation. Express irrational solutions in exact form.
36. 86+3x = 4
37. 3*+* = V3
38. log, 64 = -3
39, 5* = 3*+2
40. 252x = 5-12
41. log3 Vx – 2 = 2
42. 8 = 4. 25x
43. 2 .5 = 10"
44. log, (x + 3) + log, (x + 4) = 1
45. el-x = 5
46. 9* + 4.3* - 3 = 0
47. Suppose that f(x) = log2(x – 2) + 1.
(a) Graph f.
(b) What is f(6)? What point is on the graph of f ?
(c) Solve f(x) = 4. What point is on the graph of f?
(d) Based on the graph drawn in part (a), solve f(x) > 0.
(e) Find f (x). Graph f on the same Cartesian plane
49. Limiting Magnitude of a Telescope A telescope is limile
in its usefulness by the brightness of the star that it is ail
at and by the diameter of its lens One measure of a stai s
brightness is its magnitude: the dimmer the star, the lalge
its magnitude. A formula for the limiting magnitude L
telescope-that is, the magnitude of the dimmest star tie
can be used to view-is given by
as f.
48. Amplifying Sound An amplifier's power output P (in watts)
is related to its decibel voltage gain d by the formula
L = 9 + 5.1 log d
where d is the diameter (in inches) of the lens.
P = 25e0.ld
(a) Find the power output for a decibel voltage gain
of 4 decibels.
(b) For a power output of 50 watts, what is the decibel
voltage gain?
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
