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What interest rate (compounded annually) would you need to earn in order to double your investment in 15 years?

Compound interest is exponential.
Use the general equation where
initial investment,
number of years, and
total investment.

If you invest $100 initially, you will have $200 when it has doubled.
(You can start with any initial value you want).
Input the values into the equation.

Divide both sides by 100 and take the 15th root of both sides

(or raise both sides to the  power.)

Convert the multiplier to an interest rate, I.
Since this is an increasing exponential function, subtract 1 before converting to a percent.