Continuous compounding

Problem

A sum of $5000$ euros is invested at $3\%$ under continuous compounding. After how many years does it double?

Solution

Under a continuous regime the capital evolves according to $C(t) = C_0\,e^{rt}$ with $r = 0{,}03$. Doubling means $C(t) = 2C_0$, that is $e^{0{,}03\,t} = 2.$ Applying the natural logarithm to both sides (the taking-logarithms method): $0{,}03\,t = \ln 2 \implies t = \frac{\ln 2}{0{,}03} \approx \frac{0{,}693}{0{,}03} \approx \boxed{23{,}1 \text{ years}}.$ The doubling time does not depend on the initial capital.

Links

Topics: Exponential function
Concepts: Continuous compounding · Compound interest · Euler’s number
Functions: Exponential function
Skills: Modelling · Solving equations
Exercise type: Modelling problem