Refer to the following in doing Exercises 37 and 38. For small interest rates, there is little difference between compounding annually, quarterly, monthly, daily, or continuously. Investigating doubling times with continuous compounding leads to understanding why the rule of 72 in Exercise 11 (page 899) works. Recall that for continuous compounding of an initial principal of at annual rate , the balance at the end of years is . For the initial principal to double, we have , so . Taking the natural logarithm of both sides yields , where ln stands for the natural logarithm, represented on a calculator by a button marked either or —not or (those stand for a different kind of logarithm). Using the button gives ln . So we have , from which we can determine if we know .