Exponential and Logarithmic Functions
Three of the most common applications of exponential and logarithmic functions have to do with interest earned on an investment, population growth, and carbon dating.
When the interest earned on an investment is simple, the investor only earns interest on his initial investment. The interest earned with simple interest is the product of the interest rate, the time since the investment (usually measured in years), and the principal. The value of the an investment with simple interest after t years can be modeled by the function A(t) = P + Prt , where P is the principal, and r is the interest rate.
A compound interest plan pays interest on interest already earned. The value of an investment depends not only on the interest rate, but how frequently the interest is compounded. If, for example, a $100 investment is made with 5% interest compounded annually, after one year, the investment will be worth $105. The next year, the interest added to the value of the investment will be 5% of the $105. Compound interest causes the amount of interest earned to increase with every compounding period.
Let A(t) model the value of an investment with compound interest. A(t) = P(1 + )nt , where P is the principal, r is the interest rate, n is the number of times the interest is compounded each year, and t is the number of years since the investment was made.
When the interest on an investment is compounded continuously, a natural exponential function is used. Let the function A(t) model the value of an investment made with continuous compounding. A(t) = Pe rt , where P is the principal, r is the interest rate, and t is the number of years since the investment was made. Continuously compounded interest allows for the fastest growth of the value of an investment.
When a population has a constant relative growth rate, its size can be calculated using a natural exponential function. The population P after t units of time P(t) = P(0)e kt , where k is the constant relative growth rate, and P(0) is the initial population, measure at time zero. The units of time used in problems like these usually are proportional to the life span of the organisms of the population. For populations of bacteria, hours or days are common, and for people, years are common. Populations can also be shrinking. In this case, the value of k is negative--everything else remains the same.
Carbon dating is a technique used by scientists to date fossils. It relies on the principle that the C 14 isotope of carbon decays (this is called radioactive decay) into another isotope of carbon, C 12 at a rate proportional to its mass. This means that it has a constant relative rate of decay. The same equation is used for this situation as for continuous compounding and population growth. With situations involving decay, the rate of growth is always negative.
The half-life of a substance is the amount of time it takes for half of that substance to decay. It is only a property of substances that decay at a rate proportional to their mass. Through research, scientists have agreed that the half-life of C 14 is approximately 5700 years. The decay constant, k , for carbon-14 can be calculated using the half-life. Using the constant relative rate of decay function, we say that the remaining amount of carbon-14 C(t) = C(0)e -kt . k is negative because C(t) decreases as t increases.
|C(5700) = C(0)e -5700k = C(0)|
|e -5700k =|
|ln e -5700k = ln|
|-5700k = ln|
|- k =|
With a little manipulation, the function C(t) = C(0)e -kt can now be simplified to C(t) = C(0) . Then, given the estimated percentage of the original amount of C 14 left in an organism, its age can be approximated.