PRECALCULUS A TEST #3 – EXPONENTIAL AND LOGARITHMIC FUNCTIONS, PRACTICE
SECTION 3.1 – Exponential Functions and Their Graphs 1)
$3,500 is invested in an account that pays 5.5% interest compounded quarterly. What is the balance after eight years, rounded to the nearest cent?
2)
$7,250 is invested in an account that pays 3.25% interest compounded daily. What is the balance after sixteen years, rounded to the nearest cent?
3)
$6,300 is invested in an account that pays 4.1% interest compounded continuously. What is the balance after twenty-two years, rounded to the nearest cent?
4)
If f ( x ) = 3x , determine a value of x so that f ( x ) > 81.
SECTION 3.2 – Logarithmic Functions and Their Graphs 5)
Write the following exponential equation as a logarithm: 25 = 32.
6)
Write the following logarithmic equation as an exponential equation: log 8 512 = 3.
7)
Evaluate log b b . This does not require the use of a calculator.
8)
Students in an economic class were given an exam and then were re-tested each month to determine how much they remembered. The average score for the group is given by the human memory model f (t ) = 78 - 8 × ln(t + 1), where t represents the time in months. What was the average score for the students after nine months? Round the result to one decimal point.
SECTION 3.3 – Properties of Logarithms 9)
Evaluate log12 168 using the change-of-base formula. Round the result to four decimal places.
10)
Rewrite the expression below as a sum, difference, and/or multiple of logarithms. Use all the properties of logarithms that apply.
( x )( y ) log 2
5
z3
11)
Rewrite the expression below as a sum, difference, and/or multiple of logarithms. Use all the properties of logarithms that apply.
(9x ) ln 4
12)
2
y3
Rewrite the expression below as the logarithm of a single quantity:
(log x + log y ) - log z 13)
Rewrite the expression below as the logarithm of a single quantity. 2 ln ( x + 1) - ( 3ln y + 2 ln z )
SECTION 3.4 – Exponential and Logarithmic Functions 14)
Solve for x without using logarithms: 43 x+1 = 16 x -1.
15)
Solve for x using logarithms: 2e x -3 - 5 = 12. Round the result to four decimal places.
16)
Solve for x using logarithms: 3x+1 = 28. Round the result to four decimal places.
17)
Solve for x: 4log( x - 2) = 7. Round the result to four decimal places.
18)
Solve for x: ln x + 3 = -1. Round the result to four decimal places.
SECTION 3.5 – Exponential and Logarithmic Models 19)
$4,000 was invested in an account that pays 3.5% interest compounded continuously. How many years will pass before this investment will quadruple? Round the result to two decimal places.
20)
$3,550 was invested in an account that pays interest compounded continuously. After thirty months, the balance was $3,807.40. What is the interest rate earned by this account, rounded to the nearest tenth of a per cent?
21)
After seven years, the balance in an account that pays 2.65% interest compounded continuously was $2,106.69. How much was originally invested in the account, rounded to the nearest cent?
22)
The population of Omaha was 618,262 people in 1990. In 2000, the population of Omaha had risen to 716,998 people. Predict the population of Omaha in the year 2020. Round the result to the nearest person.
23)
A biologist is researching a newly-discovered species of bacteria. He puts 125 bacteria into what he has determined to be a favorable growth medium. Eight hours later, he measures 150 bacteria. Assuming exponential growth, how many bacteria will he find after twenty hours? Round the result to the nearest whole number.
24)
The half-life of Cobalt-60 is 5.3 years. If a chemist places forty-eight pounds of Cobalt60 in storage, how much will remain after twelve years? Round the result to two decimal places.
25)
The half-life of Strontium-90 is 28 years. A certain amount of Strontium-90 is placed in storage. After forty-five years, it is determined that 12.67 pounds remain. How much was originally placed in storage? Round the result to two decimal places.
**********************ANSWERS********************** 1)
æ rö A = P × ç1 + ÷ è nø
n×t
2)
æ 0.055 ö A = 3500 × ç 1 + ÷ 4 ø è A = 3500 × (1.01375 )
4×8
n×t
æ 0.0325 ö A = 7250 × ç 1 + ÷ 365 ø è
365×16
5840
æ 0.0325 ö A = 7250 × ç 1 + ÷ 365 ø è A = 12194.41817 = $12,194.42
32
A = 5418.209524 = $5,418.21 3)
æ rö A = P × ç1 + ÷ è nø
A = P × e r ×t A = 6300 × e0.041×22
4)
34 = 81 If 3x > 81, then x must be greater than 4.
6)
log base answer = exponent
A = 6300 × e0.902 A = 15526.52161 = $15,526.52 5)
log base answer = exponent 2 = 32 ® log 2 32 = 5
log8 512 = 3 ® 83 = 512
5
7)
logb b n = n
8) 1 2
log b b = log b b =
1 2
f ( t ) = 78 - 8 × ln ( t + 1) f ( 9 ) = 78 - 8 × ln ( 9 + 1) = 78 - 8 × ln (10 ) f ( 9 ) = 78 - 8 × ( 2.302585093 ) f ( 9 ) = 78 - 18.42068074 = 59.57931926 Average score after 9 months = 59.6
9)
log12 168 =
( x ) ( y ) = log x log 2
5
10)
ln168 log168 OR = 2.062034797 » 2.0620 ln12 log12
z3
1 5
× y2 z3
1 5
log
æ 15 2 ö x × y2 a 3 = log ç x × y ÷ - log z ¬ log = log a - log b z3 b è ø
1 æ 1 ö log ç x 5 × y 2 ÷ - log z 3 = log x 5 + log y 2 - log z 3 ¬ log a × b = log a + log b è ø 1 1 log x 5 + log y 2 - log z 3 = log x + 2log y - 3log z ¬ log a n = n × log a 5
11)
ln
(9x) 4
ln
2
= ln
y3
(9x) y
(9 x )
2
3
y4 2
3
= ln ( 9 x ) - ln y 4 ¬ ln 2
3 4
a = ln a - ln b b
3 3 2 ln ( 9 x ) - ln y 4 = 2ln ( 9 x ) - ln y ¬ ln a n = n × ln a 4 3 3 2ln ( 9 x ) - ln y = 2 ( ln 9 + ln x ) - ln y ¬ ln a × b = ln a + ln b 4 4 3 3 2 ( ln 9 + ln x ) - ln y = 2ln 9 + 2ln x - ln y ¬ Distribute 2 through parentheses 4 4
12)
( log x + log y ) - log z = log xy - log z ¬ log a × b = log a + log b log xy - log z = log
13)
xy a ¬ log = log a - log b z b
2ln ( x + 1) - ( 3ln y + 2ln z ) = ln ( x + 1) - ( ln y 3 + ln z 2 ) ¬ ln a n = n × ln a 2
ln ( x + 1) - ( ln y 3 + ln z 2 ) = ln ( x + 1) - ln y 3 z 2 ¬ ln a × b = ln a + ln b 2
2
ln ( x + 1) - ln y z 2
14)
3 2
( x + 1) = ln y z
43 x+1 = 16 x -1 43 x +1 = ( 42 )
3 2
2
¬ ln
a = ln a - ln b b
15)
x -1
2e x -3 = 17
43 x +1 = 42 x - 2 3x + 1 = 2 x - 2 x = -3
16)
3x+1 = 28
2e x -3 - 5 = 12 e x -3 = 8.5 ln e x -3 = ln 8.5 x - 3 = 2.140066163 x = 5.140066163 » 5.1401
17)
4log ( x - 2 ) = 7
ln 3x+1 = ln 28
log ( x - 2 ) = 1.75
( x + 1) × ln 3 = ln 28
10log ( x- 2) = 101.75
ln 28 ln 3 x + 1 = 3.033103256 x = 2.033103256 » 2.0331
x - 2 = 56.23413252
x +1=
x = 58.23413252 » 58.2341
18)
ln x + 3 = -1
19)
1
A = P × e r ×t
ln ( x + 3) 2 = -1
16000 = 4000 × e0.035×t
1 × ln ( x + 3) = -1 2 ln ( x + 3) = -2
4 = e0.035t ln 4 = ln e0.035t
eln ( x +3) = e -2 x + 3 = 0.1353352832 x = -2.864664717 » -2.8647
1.386294361 = 0.035t 39.60841032 = t t = 39.61 years
20)
A = P × e r ×t 3807.40 = 3550 × e r ×2.5 1.072507042 = e 2.5r ln1.072507042 = ln e 2.5 r 0.069998938 = 2.5r 0.0279995752 = r r » 2.8%
21)
A = P × e r ×t 2106.69 = P × e 0.0265×7 2106.69 = P × e 0.1855 2106.69 = P × 1.2038202 2106.69 = P × 1.2038202 1750.003863 = P P » $1,750.00
22)
y = a × e k ×t
23)
y = a × e k ×t
24)
716998 = 618262 × e k ×10 1.159699286 = e10 k ln1.159699286 = ln e10 k 0.1481607354 = 10k 0.0148160735 = k y = 618262 × e0.0148160735×30
150 = 125 × e k ×8 1.2 = e8 k ln1.2 = ln e8 k 0.1823215568 = 8k 0.0227901946 = k y = 125 × e 0.0227901946×20
y = 618262 × e 0.4444822061 y = 964292.3554 » 964,292 people
y = 125 × e 0.455803892 y = 197.1801207 » 197 bacteria
y = a × e k ×t
25)
y = a × e k ×t
0.5 = 1 × e k ×5.3 0.5 = e5.3 k ln 0.5 = ln e5.3k -0.6931471806 = 5.3k -0.1307824869 = k y = 48 × e-0.1307824869×12
0.5 = 1 × e k ×28 0.5 = e 28 k ln 0.5 = ln e 28 k -0.6931471806 = 28k -0.0247552564 = k 12.67 = a × e -0.0247552564×45
y = 48 × e -1.569389843 y = 9.992263745 » 9.99 pounds
12.67 = a × e -1.11398654 12.67 = a × 0.3282477763 38.59889058 = a a » 38.60 pounds
