Problem: What is the final difference in account values if $100,000 is invested in a) a project with returns of 7% / yr or b) one with returns of 5% / yr. Let the period of investment be 40 years. This problem is quite simple in theory, and I know the answer should be 100,000(1.07^40 - 1.05^40) = $793446.91. However, this problem is for a logic class in which we are not permitted to use calculators. Using a couple Taylor expansion approximations -- for the range of values in question, (1+x)^n = e^(nx) -- I get the following: 1.07^40 = e^2.8 and 1.05^40 = e^2. Basically, all I need to know is how to simplify the expression e^2.8 - e^2 , and then I can simply multiply by $100k to get an accurate approximation.