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LOGARITHMS AND EXPONENTS
Logarithms & Exponential Growth and Decay Review
Worksheet #16
Simplify:
1.
log 3 9
2.
 1 
log 6  
 36 
3.
ln e10
4.
log100
5.
log 7 7
6.
6log6 5
7.
ln
1
e3
8.
e 2ln 7
9.
8log2 3
Write as a single logarithm
log 3 x  log 3 5
10.
12.
3
log 4  log 5
2
11.
ln x  ln 8
13.
ln x  2 ln y  3ln z
Given log 7 2  a , log 7 3  b and log 7 5  c , in terms of a, b and c, find:
14.
log 7 10
15.
log 7 15
16.
log 7 8
17.
log 7 3 75
18.
log 7 98
19.
log 7
20.
log 7  21
21.
log 5 6
22.
log14 27
2
2
2
Solve for x, giving exact answers:
23.
log 4 x  3
24.
log(2 x  1)  log 3  log12
25.
2 log 6 x  log 6 5  1
26.
5  log 3 4  log 3 ( x  3)
27.
ln x  7
28.
2 log 3 (4 x)  0
29.
ln x 2  4
30.
log15 x  log15 ( x  2)  1
31.
7 2 x  3(7 x )  10  0
32.
e 2 x  6e x  8  0
33.
e4 x  20
34.
52 x 1  26(5x )  5  0
35.
2 x  22 x  3  0
36.
e x  10e x  3  0
Solve for x, rounding answers to 4 significant digits:
37.
4 x  20
38.
3x  5x  2
39.
62 x 3  27
40.
82 x 1  11x 8
41.
In a neighborhood where property values are expected to appreciate, a house is
purchased at a cost of $87,000. In order to be able to sell the house in 15 years and
make a 150% profit, what would the annual rate of appreciation need to be?
42.
A yeast culture doubles in size every 3 hours. If the yeast culture is now 20 cc, what
will it be in 24 hours?
43.
The population of a country increased from 150 million to 400 million in 75 years.
How long will it take, to the nearest year, for the population to reach 600 million (from
the year it was 400 million), if the rate of increase remains the same?
44.
If the annual inflation rate is 5.4%, what was the rate of increase for any given
month?
45.
If the cost of an item is increasing at a rate of 6% per year, about how long will it take
for the price to double? How long to triple?
46.
Tritium, a radioactive isotope of hydrogen, was a half-life of 12.4 years. Of an initial
sample of 100 grams, how much will remain after 60 years?
47.
One of the isotopes of plutonium, 237 Pu , decays with a half-life of 40 days. How
much of an initial sample will remain after 90 days?
48.
The half-life of one isotope of californium, 253 Cf , is 18 days. The half-life of another,
254
Cf , is 60 days. If 1 gram of each is present initially, how much of 254 Cf will be
present when only 1/4 gram of 253 Cf remains?
49.
If the population of a country is approximately 103 million, with an annual birth rate of
23 per 1000 and an annual death rate of 12 per 1000, what will the population be in
50 years?
50.
The spread of ungulate fever through a herd of cattle has a rate of 27% new
infections per day. A rancher finds 2 cases of the fever in his herd. If left untreated,
how many cattle will be infected in 1 week?
51.
Iodine-131 is a radioactive material that decays according to the equation
A = A o e 0.087 t . What is the half-life of Iodine-131?
52.
How long will it take for 100 grams of Iodine-131 to decay to 10 grams?
53.
A sample of bacteria contains 500 bacteria. One hour later 800 are present. How
many will be present after 5 hours? How long until there are 20,000 bacteria?
54.
When a certain drug enters the bloodstream, it gradually dilutes, decreasing
exponentially with a half-life of 10 hours. If the initial dose is 20 mg, how much is in
the bloodstream 6 hours later?
55.
What is the hourly rate of decrease in the amount of the drug in the bloodstream?
56.
The biological half-life of the asthma medication theophylline is 4.5 hours for
smokers. Twelve hours after a dose of 400 mg is administered, find the amount of
the drug retained in a smoker’s system.
57.
What is the hourly rate of decrease for theophylline in a smoker?
58.
For a non-smoker, the half-life of theophylline is 8 hours. Find the amount of the 400
mg dose in a non-smokers system after 12 hours.
59.
What is the hourly rate of decrease for theophylline in a non-smoker?