Modeling with Exponential and Logarithmic Functions

MATH 101
Lia Vas
Modeling with Exponential and Logarithmic
Functions
Recall that a linear function has constant average rate of change: difference of two
y-values for equally spaced x-values is constant.
Exponential function has constant percent (relative) rate of change: quotient of two
y-values for equally spaced x-values is constant.
Exponential function:
P = P 0 a t
where P 0 is the initial quantity, t is independent variable (e.g. time), and a = 1 + r is the
growth (a > 1) or decay (0 < a < 1) factor and r is the discrete (e.g. annual) percent
rate of change. For example, this formula is used to model the population growth with
given annual percent rate given.
If k is the continuous percent rate of change, then the function
P = P 0 e kt
models the change of desired quantity. Connection: if k = ln(1 + r) ≈ r if r is small.
If k is positive it is called the growth factor. If k is negative, then the function is
decreasing and k is called the decay factor. For example, this formula is used to model
the decay of a radioactive substance.
Problems:
1. The time T in minutes for a small plane to climb to an altitude of h feet is given by
T (h) = 50 log( 20000 ) where 20000 is the plane’s absolute ceiling. How long would
20000−h
it take the plane to climb to an altitude of 5000 feet Can the plane ever reach the
absolute ceiling How high is the plane 50 minutes after the take of
2. The amount of the drug (in milligrams) present in patients bloodstream t hours after
it has been administered can be modeled by A(t) = 5.5e −.234t , How many milligrams
are initially administered After how many hours will there be 1 mg of the drug in
the bloodstream
3. The pH of a liquid is defined as pH = − log[H + ], where [H + ] is the hydrogen ion
concentration in moles per liter. Find the hydrogen ion concentration for a pH of
5.2. If 9.98 · 10 −8 is the the hydrogen ion concentration for blood, find the pH.
4. Suppose that the spread of a flu at a school with 2000 students total is given by
2000
f(t) = where t represent the number of days and f represents the number
1+1999e −.8t
of students infected. How many students are initially infected How many students
are infected after a week After how many days will 45% of the total number of
students be infected

5. A bacteria culture grows by the exponential model y = 200e kt . How many bacteria
are there initially If the number of bacteria triples in 2 hours, find the number of
bacteria after 5 hours.
6. A bacteria culture starts with 600 bacteria and grows by the exponential model
y = y 0 e kt . After 3 hours there are 2400 bacteria. Find the number of bacteria after
4 hours. When will the number of bacteria be 5000
7. Suppose that a population grows by 3% per year. Find the time it would take for
the population to double.
8. Suppose that a population grows by 5% per year. Find the time it would take for
the population to triple.
9. Happyville and Smileytown both have a population of 10,000 people presently. Happyville
is increasing by 1500 people a year and Smileytown is increasing by 15% a
year.
a) Which town is growing faster
b) Find formulas for the populations of these towns as function of time t in years.
c) Use part b) to predict the size of both towns 5 years from now.
d) Find a year in which population of Happyville will be over 25000. Do the same
for Smileytown.
10. The half-life of bismuth 210 is 5 days. How many days it will take the 1.5 grams of
bismuth 210 to decay to 0.3 grams
11. The biological half-life of triazolam, a drug used for treating insomnia, is 2.3 hours.
What percent of an initial dose will remain after 5 hours
12. The table below shows the number of rabbits in a local forest from 1982 to 1996.
year 1982 1985 1988 1991 1994 1996
number of rabbits 20 67 139 182 196 198
Find a logistic model to fit this data. In what year were there 99 rabbits
13. The table below shows the concentration of a drug in a patients bloodstream t hours
after it was administered.
time (hours) 0 1 2 3 4 5
concentration (mg/cc) 2.5 2.29 2.1 1.95 1.81 1.7
Find the exponential model that fit this data. When will the concentration drop
bellow 1.2 mg/cc
14. A new drug was put on the market in 1990. The table below shows the number of
prescriptions written for this drug over a 10 year period.
year 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999
numb. of prescr. 142 149 154 155 159 161 163 164 164 166
Find a logarithmic model for this data. Using the model, how many prescriptions
will be written in 2006 In what year will there be 178 prescriptions

15. The table below shows the yield (in mg) of a chemical reaction in the first 6 minutes.
time (minutes) 1 2 3 4 5 6
yield (mg) 1.2 6.9 9.3 12.7 14.1 15.7
Use the scatterplot to find the best model to fit this data.
determine in how many minutes will the yield be 20mg.
Using that model,
16. The table below shows the number of bacteria present in a culture t hours after the
start of the experiment.
time (hours) 0 1 2 3 4 5
number of bacteria 250 342 521 736 1015 1483
Use the scatterplot to find the best model to fit this data. According to that model,
how many bacteria will be in the culture in 7 hours When will there be 5000
bacteria
Solutions:
1. 6.25 minutes. No (zero in the denominator). 18000 feet.
2. 5.5 milligrams. 7.3 hours
3. 6.31 · 10 −6 moles per liter. pH is 7.4.
4. Initially, 1 student; week later, 238; 9.25 days after, 45% of 2000 are infected.
5. 200 bacteria. 3118 bacteria. 6. 3810 bacteria. 5000 bacteria in 4.6 hours.
7. P = P 0 (1.03) t . t = 23.45 years. 8. P = P 0 (1.05) t . t = 22.52 years.
9. a) Smileytown; b) y 1 = 10000+1500t, y 2 = 10000(1.15) t ; c) 17500 and 20114 people;
d) t = 10 years and t = 6.556 ≈ 6.5 years.
10. 11.6 days 11. 22.16% 12. In 1986.
13. 9.36 hours (approximately 9 hours and 20 minutes)
14. 172 prescriptions. In year 2021. 15. Logarithmic model. 10.34 min.
16. Exponential model. 3025 bacteria. 8.4 hours (8 hours 24 minutes).