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Ch4 practice test
1.Find the exponential function f (x) = ax whose graph is given.
2.Find the function of the form f (x) = Cax whose graph is given.
3.Radioactive iodine is used by doctors as a tracer in diagnosing certain thyroid gland disorders.
This type of iodine decays in such a way that the mass remaining after t days is given by the
function below where m (t) is measured in grams.
m (t) = 2e-0.082t
(a) Find the mass at time t = 0.
(b) How much of the mass remains after 5 days? (Round your answer to 2 decimal places.)
4.An investment of $1000 is deposited into an account in which interest is compounded as given for 5
years. Complete the table by filling in the amounts to which the investment grows at the the rate of 6%.
(Round your answer to the nearest cent.)
(a) quarterly
(b) daily
5.Express the equation in exponential form ay = x.
(a)
(b)
(c)
6.Evaluate the expressions.
(a) log7(77)
(b) log7(343)
(c) log6(6)
(d)log5 (1/3125)
(e) log2 √2
(f) log10 0.1
(g)
(h)
(i)
(j)
(k)
(l)
7.If $16,000 is invested at an interest rate of 10% per year, compounded monthly, find the value
of the investment after the given number of years. (Round all answers to the nearest cent.)
(a) 5 years
(b) 10 years
8.The present value of a sum of money is the amount that must be invested now, at a given rate
of interest, to produce the desired sum at a later date.
Find the present value of an investment if the desired sum is $10,000 and interest is paid at a rate
of 8% per year, compounded quarterly, for 4 years. (Round the answer to the nearest cent.)
9.Use the definition of the logarithmic function to find x.
(a) logx(3) = 1/2
(b) logx(5) = 1/3
10.Sketch the graph the function, not by plotting points. State the domain, range, asymptote, and
intercepts if any.
y   log 2 ( x  2)  2
g (x) = log2(x + 2) – 2
y  2 x  2  2
y  2 x 2  2
11.A function f(x) is given.
f(x) = log2(log7(x))
f (x) = ln(ln(ln 3x))
(a) Find the domain of the function f.
(b) Find the inverse function of f.
12.Evaluate the expression.
log3 (21) - log3 (70) + log3 (270)
.
ln(ln ee300)
log2(846)
13.Use the Laws of Logarithms to expand the expression.
14.Use the Laws of Logarithms to combine the expression.
15.Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six
decimal places. log4 99
16.Simplify the following: (log8 7)  (log7 9)
17.Find the solution of the exponential equation, correct to four decimal places.
e4x = 11
43 - x = 19
3x = 4x + 7
18.Solve the equation for x.
x28x - 100(8x) = 0
e2x - ex - 42 = 0
19.Solve the logarithmic equation for x.
2 log(x) = log(2) + log(5x - 8)
log7 x + log7 (x - 6) = 1
(log x )3 = 7 log x
20.A man invests $5800 in an account that pays 6% interest per year, compounded continuously.
How long will it take for the amount to be $9000? (Round your answer to one decimal.)
21.A sum of $1000 was invested for 4 years, and the interest was compounded semiannually. If this sum
amounted to $1440.71 in the given time, what was the interest rate? (Give your answer as a percent
and correct to 2 decimal places.)
22.The number of bacteria in a culture is modeled by the function below where t is measured in
hours: n(t) = 400e0.3t
(a) What is the initial number of bacteria?
(b) What is the relative rate of growth of this bacterium population? Express your answer as a
percentage.
(c) How many bacteria are in the culture after 5 hours?
(d) After how many hours will the number of bacteria reach 30,000?
23.The fox population in a certain region has a relative growth rate of 6% per year. It is estimated that
the population in 2000 was 23,000.
(a) Find a function that models the population t years after 2000.
(b) Use the function from part (a) to estimate the fox population in the year 2008. (Round the
answer to the nearest whole number.)
24.A culture contains 1200 bacteria initially and doubles every 30 minutes.
(a) Find a function that models the number of bacteria n(t) after t minutes. (Enter the growth rate
using natural logarithm.)
n(t) =
(b) Find the number of bacteria after 8 hours.
(c) After how many minutes will the culture contain 7000 bacteria? (Round your answer to one
decimal place.)
25.The count in a culture of bacteria was 200 after 2 hours and 12,800 after 6 hours.
(a) What is the relative rate of growth of the bacteria population? Express your answer as a
percentage. (Round your answer to the nearest whole number.)
(b) What was the initial size of the culture? (Round your answer to the nearest whole
number.)
(c) Find a function that models the number of bacteria n (t) after t hours.
(d) Find the number of bacteria after 4.5 hours. (Round your answer to the nearest whole
number.)
(e) When will the number of bacteria be 25,000? (Round your answer to 2 decimal places.)
26.The half-life of radium-226 is 1600 years. Suppose we have a 22 mg sample.
(a) Find a function that models the mass remaining after t years.
(b) How much of the sample will remain after 3500 years? (Round your answer to the nearest
whole number.)
(c) After how long will only 18 mg of the sample remain? (Round your answer to the nearest
whole number.)
27. The mass m(t) remaining after t days from a 50 g sample of thorium-234 is given by the
following formula.
m(t) = 50e-0.0277t
(a) How much of the sample will remain after 20 days? (Round your answer to the nearest tenth.)
(b) After how long will only 30 g of the sample remain? (Round your answer to the nearest tenth.)
(c) Find the half-life of thorium-234. (Round your answer to the nearest tenth.)
28.Radium-221 has a half-life of 30 seconds. How long will it take for 94% of a sample to decay? (Round
your answer to the nearest tenth.)
29.If 250 mg of a radioactive element decays to 230 mg in 36 hours, find the half-life of the element.
30.The burial cloth of an Egyptian mummy is estimated to contain 60% of the carbon-14 it contained
originally. How long ago was the mummy buried? (The half-life of carbon-14 is 5730 years.)
31.An unknown substance has the following hydrogen ion concentration.
[H+] = 3.80 10-8 M
Find the pH. (Round your answer to one decimal place.)
Classify the substance as acidic, basic or neutral.
32.The pH reading of a glass of liquid is given. Find the hydrogen ion concentration of the liquid.
(Give your answers in scientific notation, correct to three significant numbers.)
(a) Beer: pH = 4.3
(b) Water: pH = 6.8
33.If one earthquake is 190 times as intense as another, how much larger is its magnitude on the Richter
scale?
34.Earthquake A had a magnitude of 6.9 on the Richter scale. Earthquake B had a magnitude of 7.1 on
the Richter scale. How many times more intense was earthquake B than earthquake A? (Round your
answer to two decimal places.)
35.The intensity of the sound of traffic at a busy intersection was measured at 9.00
the intensity level in decibels.
10-2 W/m2. Find
36The noise from a power mower was measured at 110. The noise level at a rock concert was measured
at 116. Find the ratio of the intensity of the rock music to that of the power mower.
37.A culture contains 20,000 bacteria initially. After an hour the bacteria count is 32321.
(a) Find the doubling period.
(b) Find the number of bacteria after 3 hours.