Natural Log Calculator
Calculate the natural logarithm ln(x) and antilog eˣ with step-by-step explanation.
Results
What Is the Natural Logarithm?
The natural logarithm ln(x) is the logarithm with base e, where e ≈ 2.71828... (Euler's number). It answers: "To what power must I raise e to get x?" ln(e) = 1, ln(1) = 0, ln(e²) = 2. The natural log is fundamental in calculus, physics, economics, and probability theory.
Euler's Number (e)
e is an irrational mathematical constant approximately equal to 2.71828182845. It is the base of natural logarithms and appears naturally in calculus (as the derivative of eˣ is eˣ itself), compound interest (continuous compounding: A = Pe^(rt)), and many probability distributions.
Natural Log vs. Common Log
ln(x) uses base e; log(x) uses base 10. They are related by: ln(x) = log(x) / log(e) ≈ log(x) / 0.4343. The change of base formula lets you convert between them.
Natural Log Properties
ln(a × b) = ln(a) + ln(b), ln(a/b) = ln(a) − ln(b), ln(aⁿ) = n × ln(a), ln(e) = 1, ln(1) = 0. The antilog of ln is e^x: if ln(x) = y, then x = e^y.
Applications
Natural logs appear in exponential growth/decay models, half-life calculations (t = −ln(0.5)/k), information theory (entropy), finance (continuous compounding), and the normal distribution formula.
Worked Example: Half-Life Calculation
Carbon-14 has a decay constant k = 0.0001209 per year. If a sample currently contains 40% of its original carbon-14, how old is it?
Using A = A₀ × e^(−kt): 0.40 = e^(−0.0001209 × t). Take the natural log of both sides: ln(0.40) = −0.0001209 × t. Since ln(0.40) ≈ −0.9163: t = 0.9163 / 0.0001209 ≈ 7,579 years.
Key Natural Log Values
| x | ln(x) | Note |
|---|---|---|
| 1 | 0 | e⁰ = 1 |
| e ≈ 2.718 | 1 | e¹ = e (definition) |
| 2 | ≈ 0.6931 | Doubling time: t = ln(2)/growth rate |
| 10 | ≈ 2.3026 | ln(10) = 1/log₁₀(e) |
| 0.5 | ≈ −0.6931 | ln(0.5) = −ln(2) |
| 100 | ≈ 4.6052 | ln(100) = 2×ln(10) |
Frequently Asked Questions
ln(1) = 0. Any number raised to the 0 power equals 1, so e⁰ = 1, meaning ln(1) = 0.
ln(e) = 1, because e¹ = e. The natural log of Euler's number is exactly 1.
e ≈ 2.71828... It is an irrational constant, the base of natural logarithms. It appears in compound interest, calculus, and exponential growth.
ln uses base e ≈ 2.718; log uses base 10. They are related: ln(x) = log(x) / log(e) ≈ 2.3026 × log(x).
Continuous compound interest: A = P × e^(rt), where ln(A/P) = rt. Also used in bond pricing, options pricing (Black-Scholes), and calculating continuous growth rates.
The rule of 70 estimates how long it takes something to double: years to double ≈ 70 / growth rate %. This comes from the exact formula: t = ln(2) / r ≈ 0.6931 / r. At 5% growth: exact time = ln(2)/0.05 ≈ 13.86 years; rule of 70 gives 70/5 = 14 years. The approximation works because 0.70 ≈ ln(2) × 1.01 when r is small.
The natural log has a uniquely simple derivative: d/dx[ln(x)] = 1/x. This makes it essential for integrating 1/x, solving differential equations, and simplifying products using logarithm rules. The integral of ln(x) is x·ln(x) − x + C. No other logarithm base has such a clean calculus form, which is why e and ln appear throughout higher mathematics and physics.