37. Solve log x = log (x 2 - 12). Explain your answer.
38. Solve 5 2x = 100 to the nearest hundredth.
39. Solve 2 x + 2 = 64 using more than one method.
40. Photography On many cameras, the amount of light admitted through the lens
can be controlled by changing the size of the opening, or aperture. The size of the
aperture is measured as an f-stop setting. The relationship between the f-stop and
the amount of light admitted can be represented by the equation n = log 2__1 , where
n is the change in f-stop setting from the starting value, f/5.6.
F-stop Setting
f/2
f/2.8
f/4
f/5.6
f/8
f/11
f/16
Change in F-stop Setting
ⴚ3
ⴚ2
ⴚ1
0
1
2
3
a. Solve the equation for when the f-stop setting is increased to f/16.
b. Solve the equation for n when the light admitted through the lens is
twice the amount at f/5.6. What is the f-stop setting? Use a calculator
to verify the solution.
Music
41. Music The frequency of a note on the piano, in Hz, is related to its position on the
n
_
keyboard by the function f (n) = 440 · 2 12 , where n is the number of keys above or
below the note concert A. (A negative value for n means that the key is to the left of,
or lower on the keyboard than, concert A.) Find the position n of the key that has a
frequency of 110 Hz.
Pianos could be
considered both
percussion and stringed
instruments, with
hammers that hit each
string. Pianos are usually
located near the drums
in an orchestra.
42. Finance Suppose that $250 is deposited into an account that pays 4.5%
n
compounded quarterly. The equation A = P 1 + __4r gives the amount A in the
account after n quarters for an initial investment P that earns interest at a rate r.
Solve for n to find how long it will take for the account to contain at least $500. (Hint:
Divide both sides by P first.)
(
)
43. Critical Thinking How many real-number solutions are there for log x 2 < 2 log x?
Use a calculator to graph and verify the answer. Explain what the graph indicates
about the answer.
44.
/////ERROR ANALYSIS/////
When a student solved log x + 4 = 8, he arrived at
99,999,996. Give a possible reason for the error.
45. Write About It Describe two methods you can use to solve an exponential
equation. Give an example of when you would use each method.
46. This problem will prepare you for the Multi-Step Test Prep on page 552.
t
The number of farms in Iowa (in thousands) can be modeled by N(t) = 119(0.987) ,
where t is the number of years since 1980.
a. Has the number of farms in Iowa been increasing or decreasing since 1980?
How can you tell?
b. Find the number of farms in Iowa in 1980 and 2000.
c. According to the model, when will be the number of farms in Iowa be
about 80,000?
7- 5 Exponential and Logarithmic Equations and Inequalities
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