The future value (FV) of an investment using compound interest is given by the equation $\mathrm{FV}\=\mathrm{PV}{\left(1\+r\right)}^{t}$, where PV is the present value of the investment, r is the interest rate, and t is the number of compounding periods.
When applying the rule of 70, we want to find out how many years it will take for the investment to grow to twice its size, so we can let $\mathrm{PV}equals;P$, the amount of the initial or principal investment, and $\mathrm{FV}\=2P$. Also, we are assuming a compounding period of 1 year, so the number of compounding periods, t, will equal the number of years.
To find the exact doubling time, we can solve the equation $2P\=P{\left(1\+r\right)}^{t}$ for t to get $t\=\frac{\mathrm{ln}\left(2\right)}{\mathrm{ln}\left(1\+r\right)}$, then substitute the current annual interest rate for r. Noting that $\mathrm{ln}\left(2\right)\approx 0.70$ and that for small values of r (as is common for annual interest rates and other growth rates), $\mathrm{ln}\left(1plus;r\right)\approx r$, this equation becomes $t\approx \frac{0.70}{r}$.
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