natural logarithm (ln)

Also called: ln, natural log, logarithm

The power to which e ≈ 2.718 must be raised to give a number: ln(eˣ) = x. It turns powers into products, ln(aᵗ) = t·ln a, so it solves for time: aᵗ = b gives t = ln b ÷ ln a.

The natural logarithm, ln, undoes a power of e: ln(eˣ) = x, where e ≈ 2.718 is the growth factor of continuous compounding. You rarely need e itself; you need what ln does to powers:

$\ln(a^t) = t \cdot \ln a \qquad a^t = b \;\Rightarrow\; t = \frac{\ln b}{\ln a}$

That is how you find time in growth problems: how long until money doubles, until a cohort pays back, until a balance reaches a target. Adding inside ln is multiplying outside, which is why logistic models add up log-odds and the coefficients become odds multipliers eᵇ.

In a spreadsheet the function is =LN(x); =EXP(x) gives eˣ.

Example

Doubling at 25% a year: t = ln 2 ÷ ln 1.25 = 0.6931 ÷ 0.2231 ≈ 3.1 years.
Payback with churn: margin 200 a month, churn 5%, CAC 1,500: t = ln(1 − 1,500 × 0.05 ÷ 200) ÷ ln 0.95 = ln 0.625 ÷ ln 0.95 ≈ 9.2 months.

Common mistakes

ln of zero or a negative number. It does not exist; in payback formulas it means "never".
Mixing log bases. ln b ÷ ln a works with any base, but both must use the same one.
Reading eᵇ as a probability change. In logistic models it multiplies the odds.

Learn it in the course

e, ln, odds and the sigmoid · ln undoes a power, so it solves for time; odds = p ÷ (1 − p), log-odds and the sigmoid p = 1 ÷ (1 + e⁻ᶻ), eᵇ as an odds multiplier, and payback with churn.
Loans and amortization · How a payment splits into interest and principal, months to repay, a flat fee against interest on the balance, and the real annual cost of a loan as its internal rate.
Geometric series · Sums that shrink by the same factor: seed ÷ (1 − k) for referrals, LTV ≈ margin ÷ churn and lifetime 1 ÷ churn, the finite sum for annuities, capped LTV and a savings pot.