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Name: Bernie N.
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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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