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.

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Interest

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.

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Population Growth

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.