Navigation Panel: (These buttons explained below)

## The General Situation

We've seen that e is the factor by which a continually-compounding bank account will increase if under simple interest it would have doubled (increased by 100%).

What if the simple interest isn't 100%? In other words, suppose your money wouldn't have doubled under simple interest, but increased by some other factor?

Suppose for instance the simple interest is 200%. Then we can split the time period up into two halves, with the simple interest being 100% for each half (for example, if you earn 200% simple interest per year, you're earning 100% interest for each six-month period).

At the end of the first half and the beginning of the second half, the balance under compound interest will be e times the original balance.

At the end of the second half, the balance under compound interest will be e times the balance at the beginning of the second half. In other words, it will be

e times (e times original amount)
which is times the original amount. So, if the simple interest earned is 200% (2 times original amount), the final balance under compound interest is times the original amount.

This turns out to be true in general. If the simple interest earned is R times the original balance, then the final balance under compound interest is times the original balance.

This is one of the reasons why exponentials of the form occur more frequently in practice than do exponentials with other bases such as , and why one usually uses exponentials and logarithms base e rather than base 10 or any other base. (The other, more important, reason has to do with the special role that base e exponentials play in calculus).

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