Natural Log Calculator

What is the Natural Logarithm?

The natural logarithm, denoted as ln(x), is the logarithm to base e (Euler's number, approximately 2.71828). It answers the question: "To what power must e be raised to equal x?" If ln(x) = y, then e^y = x.

How to Use This Calculator

1. Select your calculation mode: Choose from Calculate ln(x), Calculate e^x, or Solve for x
2. Enter the required value: Input your number
3. View instant results: See the result with step-by-step explanation
4. Explore properties: Learn natural logarithm rules and applications

Key Formulas

Natural Logarithm Definition

• If e^y = x, then ln(x) = y
• e ≈ 2.71828182845904523536...

Euler's Number (e)

Euler's number is defined as:

• e = lim(n→∞) (1 + 1/n)^n
• e = 1 + 1/1! + 1/2! + 1/3! + 1/4! + ...

Relationship to Other Logarithms

• ln(x) = log₁₀(x) / log₁₀(e) = log₁₀(x) × 2.303
• ln(x) = log₂(x) / log₂(e) = log₂(x) × 0.693
• log₁₀(x) = ln(x) / ln(10) = ln(x) × 0.434

Natural Logarithm Properties

Product Rule

ln(xy) = ln(x) + ln(y)

Quotient Rule

ln(x/y) = ln(x) - ln(y)

Power Rule

ln(x^n) = n × ln(x)

Special Values

• ln(1) = 0 because e⁰ = 1
• ln(e) = 1 because e¹ = e
• ln(e^n) = n for any value n

Derivative and Integral

• d/dx [ln(x)] = 1/x
• ∫ ln(x) dx = x(ln(x) - 1) + C

Applications of Natural Logarithms

Continuous Growth and Decay

• Population growth: P(t) = P₀ × e^(rt)
• Radioactive decay: N(t) = N₀ × e^(-λt)
• Half-life: t½ = ln(2)/λ

Finance

• Continuous compound interest: A = P × e^(rt)
• Time to double investment: t = ln(2)/r

Physics and Engineering

• RC circuit discharge: V(t) = V₀ × e^(-t/RC)
• Newton's Law of Cooling: T(t) = Tₐ + (T₀ - Tₐ) × e^(-kt)

Statistics

• Log-normal distributions
• Maximum likelihood estimation
• Information entropy

Common Natural Logarithm Values

Note: The natural logarithm is only defined for positive real numbers. ln(0) approaches negative infinity, and ln(x) for x < 0 is undefined in the real number system.