Exponential Growth

Let's put our formula for compound interest into a slightly different form.
Right now it involves raising 1+R to the power n. Let's see if we can
express is instead as raising E (Mathematica's symbol for e, the base of the
natural logarithm) to some other power. This transformation will lead us to
further insights later in the lesson.

If you are comfortable with logarithms and exponentials, you can easily do this
transformation with pencil and paper. If you're rusty, Mathematica can help you out,
though you'll still have to provide a fair amount of guidance. Let's begin
by asking Mathematica to solve an equation:

answer = Solve[(1+R)^n == E^x, x]

In other words, let's ask Mathematica to symbolically compute the proper power of
E. (We're saving the answer in a variable so that we can use it again
below.) There's something different about the equation that we're asking Mathematica
to solve here. What is it?

The answer that Mathematica gives us doesn't look as simple as it could be, although
you won't realize that unless you are familiar with logarithms. Let's see if
Mathematica can simplify it further:

Simplify[answer]

There are any number of ways for Mathematica to give an answer, especially as answers
start becoming more complicated. Sometimes asking Mathematica to simplify a solution
will give you an expression more to your liking. At other times you might ask
Mathematica to factor an answer---and Factor is another built-in function.

Now you know all that you need to know to redefine the compound function
using a power of E. Can you do it?

In this formula, the Log[1+R] part of the exponent is called the growth
rate. In general, the exponent of E in a compound-interest calculation,
when divided through by n, yields the growth rate. The reason that we
bothered to express our compound interest formula using a power of E was just
so we could discuss growth rates.

For any interest rate, the growth rate is never larger than the interest rate.
For small interest rates (such as 5%) though, the growth rate is only a little
bit smaller than the interest rate. You can easily verify this using Mathematica.

Experiment with the new function for compound interest. It should give you the
same answers as the old formula with one exception. Just as it does with Pi,
Mathematica will not convert E into a floating point number unless you insist by
using NSolve.