Question:
why is the natural logarithmic base e used to calculate
instantaneously compounded interest and how was it discovered?
Replies:
Bernie,
When any constant is raised to the x-axis variable and the result is placed
on the y-axis (y=k^x), the result is a graph with a slope always
proportional to the value of the graph. The base e is the constant that
produces a slope EQUAL to the function. That is the mathematical definition
of the number.
It is used in compound interest because of how the interest works. At any
given moment, the rate at which money is added to the principle (dP/dt) must
equal the interest rate multiplied by the principle at that moment (rP). To
produce this, we need the function whose slope equals its value. Changing
the exponent from t (time) to rt (rate x time) changes the derivative to
rate x value. Putting in an initial principle as a multiplier does not
change proportions but does set the value at t=0 equal to the initial value.
Using base e happens to produce the simplest formula. Using base 10 would
require an additional constant in the exponent (ln 10) to make the
derivatives correct. You could also work out a more complex formula with
base r, but base e is the most convenient.
One method for calculating the value of e has been the Taylor expansion of
e^x. Since the derivative of the function is itself, every derivative is
the same function: e^x. At x=0, all these equal 1. Placing them in the
Taylor series formula is actually quite easy. The formula becomes
e^x=1+(1/1!)x + (1/2!)x^2 + (1/3!)x^3 + .... The value of e is the function
at x=1. There are other methods, but I believe this was the first and the
easiest.
Dr. Ken Mellendorf
Physics Instructor
Illinois Central College
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