What is ln 2 (Natural Logarithm of 2)?
ln 2 is the natural logarithm of 2: the power to which e must be raised to get 2. Geometrically, it equals the area under the curve y = 1/x from x = 1 to x = 2. Numerically, 2.71828… raised to the power 0.69314… equals exactly 2.
∫₁² 1/x dx = ln(2) − ln(1) = ln 2 ≈ 0.6931. This is the definition of natural log: ln(a) is the area under 1/x from 1 to a.
ln 2 is the half-life constant. Any quantity that halves at a fixed rate satisfies N(t) = N₀ · e^(-λt). The half-life is t₁/₂ = ln(2)/λ ≈ 0.693/λ. This applies to radioactive decay, drug clearance from the bloodstream, the discharge of a capacitor, and the cooling of coffee.
1 − 1/2 + 1/3 − 1/4 + ... converges to ln 2 ≈ 0.6931, oscillating around the limit. Convergence is slow: every other term overshoots.
ln 2 is transcendental (Lindemann-Weierstrass, 1885). In information theory it converts between nats and bits: 1 bit = ln(2) nats ≈ 0.693 nats. The series 1 - 1/2 + 1/3 - 1/4 + ⋯ converges to exactly ln 2. Computed: 0.69314718055994530941723212145817…
N(t) = N₀ · 2^(−t/t½) = N₀ · e^(−t·ln2/t½). ln 2 ≈ 0.693 is the decay constant. After 1 half-life: 50% remains. After 10: 0.1%.
The natural logarithm of 2 is approximately 0.69314718055994530941. It is irrational and transcendental. Ln 2 equals the area under the hyperbola y = 1/x from x = 1 to x = 2. It governs every doubling and halving: a quantity growing at rate r doubles in time ln(2)/r. In information theory, 1 bit of information equals ln 2 nats. In computing, the number of binary digits needed to represent n values is log₂(n) = ln(n)/ln(2).
Natural Log of 2 is irrational. Its decimal expansion never ends and never repeats. The digits shown below are verified against the alternating harmonic series.
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