Pre-CalcA 120
Chapter 8: Logarithmic Functions Test
A. Sprague
Name: ______________________
PART ONE: MULTIPLE CHOICE - Circle the most appropriate answer.
1. Which graph represents the inverse of
a.
?
b.
y
–5
–4
–3
–2
5
4
4
3
3
2
2
1
1
1
2
3
4
5
x
–3
–2
–3
–2
–1
–1
–2
–3
–4
–4
–5
–5
d.
5
4
4
3
3
2
2
1
1
–1
–1
1
2
3
4
5
x
–5
–4
–3
–2
–1
–1
–2
–2
–3
–3
–4
–4
–5
–5
a.
2
3
4
5
x
1
2
3
4
5
x
is
b.
c.
d.
b.
c.
d.
b. 5.33
c. 0.13
d. 8
3. Which of the following represents
a.
1
y
5
2. Another way of writing
4. Evaluate
a. 4096
–4
–3
y
–4
–5
–2
c.
–5
y
5
–1
–1
Value: 10
?
.
5. The domain of the function
a.
b.
is
c.
d.
6. Which function represents a vertical translation of 7 units down, a horizontal translation of 8 units right, a horizontal
stretch by a factor of
, no reflection in the y-axis, a vertical stretch by a factor of 6, and no reflection in the x-axis, when
compared to the base function
.
a.
b.
c.
d.
7. What is the equation for the asymptote of the function
a. x = 2
b. x = –3
c. x = –5
?
d. x = –2
8. Which of the following is equivalent to the expression
a.
b.
c.
?
d.
9. Solve
.
a. x = 6
b. x =
3
2
c. x =
2
3
d. x = 729
10. The exponential form of 𝑘 = −𝑙𝑜𝑔ℎ 5 is:
1
a. hk= 5
1
b. hk = -5
c. kh = 5
d. kh = -5
PART TWO: SHORT & LONG RESPONSE – Show all work in the space provided.
1. Solve and round your answer to two decimal places.
a)
Value:
b)
(2x – 5)log10 = (x + 4)log 7
2xlog10 – 5log10 = xlog7 + 4log7
2xlog10 – xlog7 = 5log10 + 4log7
X(2log10 – log7) = 5log10 + 4log7
X = (5log10 + 4log7) / (2log10 – log7)
X = 7.26
Xlog8 = log486
X = log486/log8
X = 2.97
2. A graduate student determines that the relationship between the length, s, in metres, of the skull and the body mass, m, in
kilograms, of a particular species can be expressed using the equation
. Determine the body
mass of an animal with a skull size of 0.56 m. Round your answer to the nearest kilogram.
3.6045log(0.56) = logm – 3.4425
3.6045log(0.56) + 3.4425 = log m
2.5348437433438494007459490422005 = log m
102.5348437433438494007459490422005 = m
M = 343
3. Solve
to the nearest hundredth.
Log(3x + 15) = log10 + log (x+3)
Log(3x+15) = log10(x+3)
3x + 15 = 10x +30
7x = -15
X = -15/7
4. The eye size of many vertebrates is related to body mass by the logarithmic equation
,
where E is the eye axial length, in millimetres, and m is the body mass, in kilograms. Predict the mass of a vertebrate with
an eye axial length of 43 mm. Round your answer to the nearest hundredth of a kilogram.
Log (43) = log 10.61 + 0.1964 log m
Log43 – log10.61 =0.1964 log m
(Log43 – log10.61)/0.1964 = log m
3.0944657417425960299501205663872 = log m
10 3.0944657417425960299501205663872 = m
1243.0 = m
5. The variable k in the function
6. The graph of
represents _____vertical translation___________.
is the graph of
translated ________6 units right____________.
4. The domain of the function 𝑓(𝑥) = 5𝑙𝑜𝑔8 3(𝑥 − 2) + 4 is _________{x | x > 2, x e R}___________.
5. The equation
6. Evaluate:
is an example of the _____power_______ law of logarithms.
a)
b)
a) 2x = 321/4
2x = 25/4
X = 5/4
7. If
40 = m – n
1=m–n
M=1+n
M = 1 + (17/2)
M = 19/2
c) log 5 40 – log5103
log 5 40/1000
log 5 0.04
Log 0.04 / log 5 = - 2
b) 6 + 3(-4)
6 – 12
-6
and
, determine the values of m and n.
42 = m + n
16 = (1 + n) + n
17 = 2n
N = 17/2
c) 𝑙𝑜𝑔5 40 − 3𝑙𝑜𝑔5 10
8. a) Sketch the graph of the function 𝑦 = 𝑙𝑜𝑔3 (𝑥 + 9) + 2.
y-int ( 0,7) x-int (-8 8/9, 0)
b) Identify the domain, the range, and the equation of the vertical asymptote.
Domain {x|x>-9, xeR}
Range {y|yeR}
X = -9
9. Solve for x. Ensure that you verify your answer(s).
a) 𝑙𝑜𝑔2 8𝑥−3 = 4
b) 𝑙𝑜𝑔2 (𝑥 2 − 2𝑥)7 = 21
24 = 8x-3
24 = 23x – 9
4 = 3x – 9
13 = 3x
X = 13/3
221 = (x2 – 2x)7
23 = x2 – 2x
8 = x2 – 2x
0 = x2 – 2x – 8
0 = (x – 4)(x + 2)
x = 4 x = -2
3
2
c) log √𝑥 2 + 48 = 3
1/3 log(x2 + 48) = 2/3
log(x2 + 48) = 2
102 = x2 + 48
100 = x2 + 48
52 = x2
√52 = x
10.The world population was approximately 6 billion in 2000. Assume that the population grows at a rate of 1.3% per year.
In what year will the population reach at least 10 billion?
10 = 6 ( 1.013)x
10/6 = 1.013x
Log 10/6 = xlog1.013
X = log(10/6)/log1.013
X = 7 years
11. Describe the effects on the graph of 𝑦 = 𝑙𝑜𝑔3 𝑥 if it is transformed to = 𝑙𝑜𝑔3 √𝑥 + 7 .
Vertical stretch of ½ , horizontal translation of 7 units left
12.
Ethanol is a high-octane renewable fuel derived from crops such as corn and wheat. Through the process of fermentation,
yeast cells duplicate in a bioreactor and convert carbohydrates into ethanol. Researchers start with a yeast-cell
concentration of 4.0 g/L in a bioreactor. 10 hours later, the yeast-cell concentration is 17.5 g/L. What is the doubling
time of the yeast cells, to the nearest tenth of an hour?
4.0 (2)10/r = 17.5
(2)10/r = 17.5/4
(10/x)log 2 = log (17.5/4)
10log 2 = x log (17.5/4)
X = (10log 2) / (log (17.5/4))
X = 4.7
BONUS:
Show/prove that
Log x3/2 + logx2 – logx1/2 = log x7/2/ x1/2 = log x3 = 3 log x
.