In Problems 36-46, solve each equation. Express irrational solutions in exact form. 36. 86+3r = 4 37. 3** = V3 38. log, 64 = 39. 5* = 3r+2 -3 40. 252 = 5r-12 41. log; Vx - 2 = 2 42. 8 = 4.25r 43. 2-5 = 10 44. log, (x + 3) + log, (x+ 4) = 1 45. el- = 5 46. 9* + 4.3* - 3 = 0
Compound Probability
Compound probability can be defined as the probability of the two events which are independent. It can be defined as the multiplication of the probability of two events that are not dependent.
Tree diagram
Probability theory is a branch of mathematics that deals with the subject of probability. Although there are many different concepts of probability, probability theory expresses the definition mathematically through a series of axioms. Usually, these axioms express probability in terms of a probability space, which assigns a measure with values ranging from 0 to 1 to a set of outcomes known as the sample space. An event is a subset of these outcomes that is described.
Conditional Probability
By definition, the term probability is expressed as a part of mathematics where the chance of an event that may either occur or not is evaluated and expressed in numerical terms. The range of the value within which probability can be expressed is between 0 and 1. The higher the chance of an event occurring, the closer is its value to be 1. If the probability of an event is 1, it means that the event will happen under all considered circumstances. Similarly, if the probability is exactly 0, then no matter the situation, the event will never occur.
![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%2Ffe0b8a70-5398-44fe-9023-cf9058a67ca8%2Fh80f36n_processed.jpeg&w=3840&q=75)
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