Natural Log Calculator | Complete ln(x) Guide

Master the natural logarithm ln(x) with Euler's number e, formulas, worked examples, and applications in calculus, physics, and finance.

The natural logarithm ln(x) is the logarithm with base e (Euler\'s number ≈ 2.71828). It is the inverse function of the exponential function e^x and appears throughout mathematics — from the derivative of 1/x to radioactive decay, compound interest, and information entropy.

Key Takeaway

The natural logarithm ln(x) is the inverse of e^x. Its derivative (1/x) makes it fundamental to calculus, and it appears in physics, finance, and information theory.

1. What Is the Natural Logarithm?

ln(x) = log_e(x), where e ≈ 2.71828

The natural logarithm answers: "e raised to what power equals x?" If e^y = x, then y = ln(x). The constant e was discovered by Jacob Bernoulli studying compound interest and later formalized by Leonhard Euler.

2. Key Properties

ln(1) = 0 (since e⁰ = 1)
ln(e) = 1 (since e¹ = e)
ln(e^x) = x (inverse property)
e^(ln(x)) = x (inverse property)
ln(x × y) = ln(x) + ln(y) (product rule)
ln(x / y) = ln(x) - ln(y) (quotient rule)
ln(x^n) = n × ln(x) (power rule)

3. Worked Examples

Example 1: ln(100)

ln(100) = log₁₀(100) / log₁₀(e) = 2 / 0.43429 ≈ 4.60517
Verify: e^4.60517 ≈ 100 —

Example 2: Solving e^x = 5

Take ln of both sides: ln(e^x) = ln(5)
x = ln(5) ≈ 1.60944
Verify: e^1.60944 ≈ 5 —

4. Applications

Field	Application
Calculus	d/dx[ln(x)] = 1/x, ∫(1/x)dx = ln\|x\| + C
Finance	Continuous compounding: A = Pe^(rt), t = ln(A/P)/r
Physics	Radioactive decay: N(t) = N₀e^(-λt), half-life = ln(2)/λ
Information	Entropy: H = -Σ p(x) ln(p(x))

The Natural Logarithm: A Masterclass on ln(x), Euler's Number, and Calculus

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