Recondite as all this may sound, the natural logarithm is extremely practical, though often inconspicuously. For one thing, it underlies a rule of thumb known to investors and bankers as the rule of 72. To estimate how long it will take to double your money at a given annual rate of return, divide 72 by the rate of return. Thus, money growing at a 6 percent annual rate doubles after about 72/6 = 12 years. This rule of thumb follows from the properties of the natural logarithm and exponential growth and works well if the interest rate is low enough.
