Unit 3 PRACTICE Test: Exponential, Logistic, and Logarithmic Functions Name_________________________________
Date_________________________
Per _________
*****YOU CAN ONLY USE A CALCULATOR FOR #1-11 ONLY!*****
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
6) A bacterial culture has an initial population of 10,000. If its population declines to 6000 in 4 hours, what will it be at the end of 6 hours? A) 2000 B) 4648 C) 2324 D) 7114
Solve the problem. 1) How long will it take for $3600 to grow to $40,200 at an interest rate of 12.1% if the interest is compounded quarterly? Round the number of years to the nearest hundredth. A) 20.24 B) 80.97 C) 21.13 D) 93.68
7) In the formula A = Iekt, A is the amount of radioactive material remaining from an initial amount I at a given time t and k is a negative constant determined by the nature of the material. A certain radioactive isotope decays at a rate of 0.225% annually. Determine the half-life of this isotope, to the nearest year. A) 134 yr B) 222 yr C) 308 yr D) 3 yr
2) How long will it take for $5500 to grow to $ 22,600 at an interest rate of 3% if the interest is compounded continuously? Round the number of years to the nearest hundredth. A) 47.11 B) 1.09 C) 4.71 D) 4710.67
8) The growth in the population of a certain rodent at a dump site fits the exponential function A(t)= 740e0.032t, where t is the
3) At what interest rate must $5700 be compounded annually to equal $8842.57 after 9 yr? (Round to the nearest percent.) A) 4% B) 5% C) 6% D) 7%
number of years since 1982. Estimate the population in the year 2000. A) 1316 B) 764 C) 1359 D) 658
4) The hydrogen potential, pH, of a substance is defined by pH = -log [H+], where [H+] is measured in moles per liter. Find the pH of a sample of lake water whose [H+ ] is 3.05 x 10-9
9) In September 1998 the population of the country of West Goma in millions was modeled by f(x) = 17.7e0.0018x. At the same
time the population of East Goma in millions was modeled by g(x) = 13.9e0.0186x. In both
moles per liter. (Round to the nearest tenth.) A) 7.3 B) 10.1 C) 6.4 D) 8.5
formulas x is the year, where x = 0 corresponds to September 1998. Assuming these trends continue, estimate the year when the population of West Goma will equal the population of East Goma. A) 14 B) 2012 C) 2010 D) 1984
5) Use the formula M = log (A/Ao), where the magnitude of an earthquake on the Richter scale is based on A, the measurement of a seismic wave, and Ao, the measurement of a
seismic wave of a level zero earthquake with the same epicenter. An earthquake was recorded which was 79,433 times more powerful than a reference level zero earthquake. What is the magnitude of this earthquake? (Round to the nearest tenth.) A) 11.3 B) 3.9 C) 4.9 D) 0.5
Use the change of base rule to find the logarithm to four decimal places. 2.9 10) log 4.1 A) 1.3252 B) 0.7546 C) 0.4624 D) 0.7073
11) log
0.968 7 A) -0.0167 C) 7.2314
1
B) -59.8315 D) -0.0141
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********************************************************************************************************************** 1 -x Solve the equation. 15) f(x) = 2 12) ex + e-x = 3
A) ln
3 3 ,ln 2 2
B) ln
-3+ 5 -3- 5 ,ln 2 2
C) ln
3+ 5 3- 5 ,ln 2 2
A) Exponential decay function;
B) Exponential growth function;
f(x) =
x
lim -
f(x)
= ; lim f(x) = 0 x
C) Exponential decay function;
D) Exponential growth function;
lim x -
Solve the equation. 16) log 3 x = log 5 + log (x - 2 ) A) -5
f(x)
= ; lim f(x) = 0 x
C) x
lim -
lim -
f(x) =
0; lim f(x) = x
D) Exponential growth function;
x
lim -
f(x) =
x
lim -
f(x)
5 4
B)
3 2
D) 5
f(x) =
; lim f(x) = 0 x x
lim -
= 0; lim f(x) = x
State whether the function is an exponential growth function or exponential decay function, and describe its end behavior using limits. 14) f(x) = e3x
A) Exponential growth function;
x
; lim f(x) = 0 x
Determine the doubling time of the investment. 13) $1400 at 6% compounded quarterly A) 9.31 years B) 11.64 years C) 23.28 years D) 17.46 years
C) Exponential decay function;
lim -
0; lim f(x) = x
3+ 5 3- 5 , D) 2 2
B) Exponential decay function;
x
f(x)
= 0; lim f(x) = x
17) log (x + 3) = 1 - log x A) -5, 2 C) -2
B) 2 D) -2, 5
18) 3(10 - 2x) = 81 A) 27 B) -3
C) 5
19) (1/4)x = 256 A) 4 C) -1/4
B) 1/4 D) -4
D) 3
Find the exponential function that satisfies the given conditions. 20) Initial mass = 463 g, halving once every 21 hours 1 21t A) m(t) = 463 · 2
B) m(t) = 463 · 221t 1 t/21 C) m(t) = 463 · 2 D) m(t) = 463 · 2t/21 2
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********************************************************************************************************************** Describe how to transform the graph of the basic function 21) Initial population = 30,427, increasing at a rate of 1.8% per year g(x) into the graph of the given function f(x). t 26) f(x) = 4 log (4 - x); g(x) = log x A) P(t) = 1.8 · 30,427 A) Reflect across the y-axis, translate 4 units B) P(t) = 30,427 · 0.018t to the left, and vertically stretch by a C) P(t) = 30,427 · 1.018t factor of 4. D) P(t) = 30,427 · 1.8t B) Reflect across the y-axis, translate 4 units to the right, and vertically stretch by a Determine a formula for the exponential function. factor of 4. 22) C) Reflect across the x-axis, translate 4 units to the right, and vertically stretch by a factor of 4. D) Reflect across the x-axis, translate 4 units to the left, and vertically stretch by a factor of 4.
A) f(x) = 4 · 2 x C) f(x) = 16 · 2 x 23)
x -2 -1 0 1 2
f(x) 4 6 9 13.5 20.25
A) f(x) = 9 · 1.33 x C) f(x) = 6 · 1.5x
27) f(x) = ln (x + 7) - 9; g(x) = ln x A) Translate 9 units to the left and 7 units up. B) Translate 7 units to the right and 9 units down. C) Translate 7 units to the left and 9 units down. D) Translate 7 units to the left and 9 units up.
B) f(x) = 4 · 16x D) f(x) = 4 · 0.5x
Rewrite the expression as a sum or difference or multiple of logarithms. 4 x 28) log y
B) f(x) = 9 · 1.5x D) f(x) = 4 · 0.5x
5 2
C) -
25) log 0.0001 A) 4 C) -1
B)
1 1 log x - log y 4 4
D)
log log
B) 5 2
log x log y
C) log
Evaluate the logarithm. 5 1 24) log3 9 A)
A)
D)
2 5
2 5
B) -10 D) -4 3
x y - log 4 4 4 4
x y
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********************************************************************************************************************** 35) 10 - log5(x + 9) = 9 5 x 29) log19 y A) x = -4 B) x = -8 C) x = 14 D) x = 4 1 A) log19 5 · log19 x ÷ log19 y 2 Find the domain of the function. 1 B) log19 y - log19 5 - log19 x x+7 2 36) f(x) = log10 x-5 C) log19 (5 x) - log19 y A) (- , -7) (5, ) B) (-7, 5) 1 D) log19 5 + log19 x - log19 y C) (- , -7) D) (5, ) 2
37) f(x) = log10 (x2 - 10x + 21)
Use the product, quotient, and power rules of logarithms to rewrite the expression as a single logarithm. Assume that all variables represent positive real numbers. 1 30) logax + 5 loga y - 2 loga x 2 y5 A) loga x3/2
C) loga
x y5
ln (7xy) ln(4yz)
C) ln
x7 y3 z4
32) 4log x + 2log y A) log(x4 + y2) C) log(8xy)
7x 4z
B) log (x4 y2 ) D) log(4x + 2y)
Find the exact solution to the equation. 33) 4 · 2x/5 = 16 2 A) x = B) x = 2 5 C) x = 9
34) 3(12 - 2x) = 81 A) x = 27 C) x = 6
B) (9, ) D) (-9, )
Graph the function. Describe its position relative to the graph of the indicated basic function. 1 39) f(x) = ln x; relative to f(x) = ln x 2
ln(x7y7 ) ln(y4z4 )
D) ln
B) (- , -3) D) (7, )
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
D) loga x4 y5
B)
(7, )
38) f(x) = ln (9 - x) A) (- , -9) C) (- , 9)
B) loga x2 y5
31) 7ln (xy) - 4ln (yz) A)
A) (-3, 7) C) (- , 3)
D) x = 10
B) x = -4 D) x = 4
4
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********************************************************************************************************************** 40) f(x) = ln x - 2; relative to f(x) = ln x 44) f(x) = log3(x + 2)
Graph the function and analyze it for domain, range, continuity, increasing or decreasing behavior, symmetry, boundedness, extrema, asymptotes, and end behavior. 45) f(x) = ln x + 2
46) f(x) = ln (3 - x) 47) f(x) = - log (x + 2) Graph the function and analyze it for domain, range, continuity, increasing or decreasing behavior, symmetry, boundedness, extrema, asymptotes, and end behavior. 48) f(x) = 4 · e -x
41) f(x) = e4x + 3; relative to f(x) = ex
49) f(x) = 4 · 3 x
42) f(x) = 2 - 4 -x; relative to f(x) = 4x
Graph the function and analyze it for domain, range, continuity, increasing or decreasing behavior, asymptotes, and end behavior. 43) f(x) = ln (x 2)
5
