Finding Doubling Time

Differential Equations · Axiom Academy

Apply the exponential growth model and natural logarithms to find when an investment doubles. An investment of 5,000 earns 6% annual interest, compounded continuously . How long will it take for the investment to double to 10,000 ? Initial investment: P_0 = 5 , 000 Growth rate: r = 0.06 (6%, continuous compounding) Target: P(t) = 10 , 000 — find t Nice work! You solved for doubling time using the exponential growth model and natural logarithms. Exponential growth model: P(t) = P_0 e^ rt describes continuous compound growth, where P_0 is the initial value, r is the growth rate, and t is time. Setting up doubling time: doubling means P(t) = 2P_0 , which cancels P_0 from both sides and simplifies the equation. Natural logarithms undo e : taking of both sides turns into rt , so t can be isolated algebraically. Universal doubling-time formula: for any continuously-compounded rate r , doubling time is — the initial amount doesn't matter. Result: at 6% continuous growth, an investment doubles in years, regardless of the starting amount. This same method — set the model equal to double the initial value, then take of both sides — works for any exponential growth scenario in differential equations: population growth, bacterial cultures, and more.

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