Natural Logarithm of Two

ln 2 = 0.693147 is the most-used logarithm in applied work because half-lives run on it: t₁⁄₂ = ln 2 / λ. A radionuclide, a drug in the bloodstream and a discharging capacitor all follow the same relation, and the reason "70" appears in the rule for doubling time — 70 divided by the percentage growth rate — is that 100 × ln 2 = 69.3, rounded up to a number with more divisors.

Its series, 1 − 1/2 + 1/3 − 1/4 + ..., is the alternating harmonic series, and it converges so slowly that it needs a thousand terms for three decimal places; it is also the standard example of a conditionally convergent series that can be rearranged to sum to any value you like, a result of Riemann's that startles every student who meets it. In information theory ln 2 is the conversion factor between nats and bits, and in computer science log₂ n = ln n / ln 2 is how a natural logarithm becomes a tree depth.