Compound interest, the number e, and natural logarithms

When interest is compounded n times a year, an amount P invested at annual rate r grows to A = P(1 + r/n)^(nt) after t years: $1,000 at 6 percent compounded monthly for 10 years becomes 1000(1.005)^120, about $1,819.40. Compounding more and more often approaches a limit called continuous compounding, A = Pe^(rt), where e, about 2.71828, is the number that (1 + 1/n)^n approaches as n grows without bound; the same $1,000 compounded continuously grows to 1000e^0.6, about $1,822.12. The natural logarithm, ln, is the logarithm with base e, and it undoes e^x. To find how long money takes to double at 6 percent compounded continuously, solve 2 = e^(0.06t): taking ln of both sides gives t = ln 2 / 0.06, about 11.55 years. Exponential functions with base e model continuous growth and decay throughout science and finance.

This explanation is AI-generated and can occasionally be wrong. 🚩 Report