Math
LN function: calculate continuous growth rates and exact doubling time
Use the natural logarithm LN (inverse of EXP) to solve for unknown compounding time or continuous log returns, and handle zero/negative #NUM! inputs.
Solve for exponent variables with LN
LN(number) returns the natural logarithm (base e) of a strictly positive number. Because LN(EXP(x)) = x, logarithms turn exponents into multiplication—letting you solve for time t in (1 + r)^t = 2 or continuous rate r in Ending = Starting * e^(r*t).
At an 8% discrete annual return, exact doubling time is LN(2) / LN(1 + 0.08) = 9.01 years (close to the mental Rule of 72 estimate 72 / 8 = 9).
| Task | Formula | Result |
|---|---|---|
| Log of 1 (no change) | =LN(1) | 0 |
| Continuous rate: 1,000 -> 1,349.86 in 5 yrs | =ROUND(LN(B2/A2)/C2, 4) | 0.06 (6.00%) |
| Doubling time at 8% annual rate | =ROUND(LN(2)/LN(1+D2), 2) | 9.01 years |
| Zero or negative input | =LN(0) | #NUM! |
Implied continuous growth rate and doubling time
Log returns are time-additive: summing annual LN(P_t / P_{t-1}) across five years equals LN(P_5 / P_0) directly.
=ROUND(LN(2) / LN(1 + D2), 2)Domain restriction: number must be > 0
No real power of e produces zero or a negative number. Passing 0 or any negative value to LN returns #NUM!.
=LN(A2)Try the example
Use the original values or change them to test your understanding.