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Calculate natural logarithm ln(x) = loge(x) where e ≈ 2.71828
Must be positive (x > 0)
The natural logarithm ln(x) is the logarithm with base e (Euler's number ≈ 2.71828). It answers: "To what power must I raise e to get x?" For example, ln(e²) = 2 because e² equals e².
e ≈ 2.71828 is a mathematical constant that appears naturally in continuous growth and decay. It's "natural" because the derivative of eˣ is eˣ, and the derivative of ln(x) is 1/x - making calculus much simpler.
ln means natural log (base e), while log typically means common log (base 10) in most contexts. In mathematics and science, ln is more fundamental. Some fields use "log" to mean natural log, so context matters.
Key properties: ln(ab) = ln(a) + ln(b), ln(a/b) = ln(a) - ln(b), ln(aᵇ) = b·ln(a), ln(1) = 0, ln(e) = 1, and e^(ln(x)) = x for x > 0.
Natural logs appear in calculus, physics, engineering, economics (continuous compound interest), chemistry (reaction rates), biology (population models), and statistics (normal distribution). They're fundamental to describing continuous processes.
No, not in real numbers. Since e raised to any real power is always positive, ln of negative numbers is undefined in real numbers. In complex numbers, ln(-1) = iπ, but that's beyond basic calculations.
Use the change of base formula: ln(x) = log₁₀(x) / log₁₀(e) ≈ log₁₀(x) / 0.43429, or log₁₀(x) = ln(x) / ln(10) ≈ ln(x) / 2.30259.
ln and the exponential function eˣ are inverses. This means ln(eˣ) = x and e^(ln(x)) = x. If y = eˣ, then x = ln(y). They "undo" each other perfectly.