What is ln(2) and what is its value?

ln(2), or the natural logarithm of 2, is the power to which e must be raised to equal 2. Its value is approximately 0.69314718056... and is a transcendental, irrational number.

Who first computed ln 2?

Nicholas Mercator published the alternating series 1 - 1/2 + 1/3 - 1/4 + ... = ln 2 in his 1668 work Logarithmotechnia. John Napier had invented logarithms in 1614, and Euler clarified the connection to base e in the 18th century.

What is the Rule of 70?

Divide 70 by a percentage growth rate to estimate doubling time. At 7% annual growth, doubling takes about 70/7 = 10 years. This works because ln 2 is approximately 0.693, so the exact doubling time is ln 2 / r, and 69.3 rounds conveniently to 70.

Why does ln 2 appear in every half-life formula?

A quantity decaying by half satisfies e^(-rate x time) = 1/2. Solving for the half-life gives t = ln 2 / rate. This is why every radioactive half-life, drug clearance rate, and RC circuit time constant contains ln 2.

Is ln 2 transcendental?

Yes. The Lindemann-Weierstrass theorem implies that if a is a nonzero algebraic number, then e^a is transcendental. Since e^(ln 2) = 2 is algebraic, ln 2 cannot itself be algebraic - so it must be transcendental.

How does ln 2 connect to information theory?

Claude Shannon's 1948 information theory paper established that one binary digit (bit) carries exactly ln 2 nats of entropy. The nat is the natural unit of information, and ln 2 is the conversion factor between bits and nats.

Christoffer De Geer March 20, 2026

What is ln 2 (Natural Logarithm of 2)?

ln 2 ≈ 0.69314…

ln 2 ≈ 0.69314718055994530941. Transcendental. Half-life = ln(2) / decay rate.

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.

Area under 1/x from 1 to 2 equals ln 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.

Alternating harmonic series converging to ln 2

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…

Euler's Number e Harmonic Series Taylor Series Riemann Zeta Function

Radioactive decay: quantity halves every half-life

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%.

Key facts about Natural Log of 2

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).

FAQ

What is ln(2) and what is its value?

ln(2), or the natural logarithm of 2, is the power to which e must be raised to equal 2. Its value is approximately 0.69314718056... and is a transcendental, irrational number.
Who first computed ln 2?

Nicholas Mercator published the alternating series 1 - 1/2 + 1/3 - 1/4 + ... = ln 2 in his 1668 work Logarithmotechnia. John Napier had invented logarithms in 1614, and Euler clarified the connection to base e in the 18th century.
What is the Rule of 70?

Divide 70 by a percentage growth rate to estimate doubling time. At 7% annual growth, doubling takes about 70/7 = 10 years. This works because ln 2 is approximately 0.693, so the exact doubling time is ln 2 / r, and 69.3 rounds conveniently to 70.
Why does ln 2 appear in every half-life formula?

A quantity decaying by half satisfies e^(-rate x time) = 1/2. Solving for the half-life gives t = ln 2 / rate. This is why every radioactive half-life, drug clearance rate, and RC circuit time constant contains ln 2.
Is ln 2 transcendental?

Yes. The Lindemann-Weierstrass theorem implies that if a is a nonzero algebraic number, then e^a is transcendental. Since e^(ln 2) = 2 is algebraic, ln 2 cannot itself be algebraic - so it must be transcendental.
How does ln 2 connect to information theory?

Claude Shannon's 1948 information theory paper established that one binary digit (bit) carries exactly ln 2 nats of entropy. The nat is the natural unit of information, and ln 2 is the conversion factor between bits and nats.

Ready to play?

Pi

Memorize pi, e, and 38 mathematical constants using the numpad path method

Play now

Works on any device.

Topic roundups

#memory All memory games Hold a sequence, grid, or list in working memory and reproduce it under pressure. #numbers All numbers games Arithmetic, ratios, units, and number sense in short timed rounds. #sequence All sequence games Reproduce or extend an ordered string of items in the right order.