Natural Log Calculator

Calculate natural logarithms (ln) and exponential functions with step-by-step solutions

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e

Euler's Number

Expression:

ln() e x where ln(x) =

Step-by-Step Solution

Natural Logarithm Properties Reference

Property Formula

Common Natural Log Values

x ln(x)

Real-World Applications

Conversion Factors

ln(10) ≈ 2.3026
ln(2) ≈ 0.6931
1/ln(10) ≈ 0.4343
1/ln(2) ≈ 1.4427

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About 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

x ln(x)
0.5 -0.693
1 0
e 1
2 0.693
3 1.099
5 1.609
10 2.303
100 4.605

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.