Download Exponential and Logarithmic Applications

Exponential Growth and Exponential Decay
The amount at time t, A(t), is given by A(t) = A0ekt where A0 is the initial population, A(0), and k
is the growth rate or decay rate.
If k > 0, the amount is growing.
If k < 0, the amount is decaying (decreasing).
Example
A population of mold spores follows the exponential growth model. The initial population was
752 and 30 days later there are 1217.
a. What function models the mold population?
b. When will the population reach 2000?
Compound Interest
The amount after t years, A(t), is given by A(t) = P(1 + r/n)nt where P is the principal in dollars, r
is the annual interest rate, and n is the number of times the interest is compounded per year.
The amount after t years, A(t), is given by A(t) = Pert when the interest is compounded
continuously.
This is an example of exponential growth.
Example
Assuming an interest rate of 3%, find the time needed to grow $5,000 to $7,000.
a. Compounding yearly.
b. Compounding monthly.
c. Compounding continuously.
Present Value
Present value is the amount of money needed today to have a certain amount in the future when
invested at an annual rate with compounding interest.
The present value, P(t), is given by P(t) = A(1 + r/n)-nt where A is the amount in dollars desired t
years in the future, r is the annual interest rate, and n is the number of times the interest is
compounded per year.
This is the compound interest formula solved for P.
The present value, P(t), is given by P(t) = Ae-rt when the interest is compounded continuously.
This is an example of exponential decay.
Example
Find the present value if $10,000 is needed in 5 years and you can find a 6% interest rate.
a. Compounding yearly.
b. Compounding monthly.
c. Compounding continuously.
Half-Life
The half-life of a radioactive material is the time necessary for half of material to decay or be
eliminated.
The amount of a radioactive material remaining, A(t) is given by A(t) = A0(1/2)t/k where A0 is the
initial amount and k is the time for half of material to decay.
This is equivalent to A(t) = A0e(-ln2/k)t.
Example
The half-life of uranium-234 is approximately 25,000 years. Initially, there are 100 grams of
uranium-234.
a. What function models the decay of this sample of uranium-234?
b. When will there be 90 grams left?
Logistic Growth Model
The logistic growth model is a more realistic growth model. It takes into account that the
population can’t continue to grow forever.
The population at time t, P(t), is given by P(t) = C/(1 + Be-rt) where C is the maximum
sustainable population and B is ratio of C - P0, which is the maximum additional population that
can be supported, and P0.
Example
Suppose a neighborhood can support a maximum of 100 deer. The current population is 57 and
five years ago it was 45.
a. What is the function that models the deer population?
b. When will the population reach 75?
Gaussian Model
2
y ae ( x b ) / c
One application is the normal distribution or the bell curve in statistics and probability.
2
2
1
y
e ( x ) /2 where μ is the mean and σ is the standard deviation.
2
Logarithmic Model
y = a + clogbx
b is usually e or 10.
Applications
Decibels, B = 10log(I/I0)
I0 = 10-12 watt/m2 – faintest sound that can be heard by the human ear
pH = -log(H+)
H+ is the hydrogen ion concentration in moles/L
Richter scale, R = log(I/I0)
I0 = 1 the minimum intensity.