Natural Log Calculator
Calculate natural logarithms (ln) and exponentials (e^x) with step-by-step solutions
Natural Logarithm Calculator
Enter a positive number to calculate ln(x)
Input must be greater than 0
Step-by-Step Solution
Natural logarithm is undefined for non-positive numbers
Common Natural Log Values
Key Properties
Domain: x > 0 (positive real numbers)
Range: All real numbers (-∞, +∞)
Base: e ≈ 2.71828 (Euler's number)
Inverse: e^x (exponential function)
Derivative: d/dx[ln(x)] = 1/x
Understanding Natural Logarithms
What is a Natural Logarithm?
The natural logarithm, denoted as ln(x), is the logarithm to the base e (Euler's number). It answers the question: "To what power must e be raised to get x?"
Mathematical Definition
ln(x) = y ⟺ e^y = x
Where e ≈ 2.71828...
Key Examples
- •ln(1) = 0 because e⁰ = 1
- •ln(e) = 1 because e¹ = e
- •ln(e²) = 2 because e² = e²
Logarithm Rules
ln(ab) = ln(a) + ln(b)
Product rule
ln(a/b) = ln(a) - ln(b)
Quotient rule
ln(a^n) = n × ln(a)
Power rule
ln(1/a) = -ln(a)
Reciprocal rule
Why "Natural"?
Natural logarithms are called "natural" because they arise naturally in calculus and mathematical analysis. The function e^x is its own derivative, making ln(x) fundamental to describing growth and decay processes in nature.
Real-World Applications
- • Population growth models
- • Radioactive decay
- • Compound interest
- • Signal processing
Mathematical Fields
- • Calculus and analysis
- • Differential equations
- • Probability theory
- • Information theory
Special Values
- • ln(2) ≈ 0.6931 (doubling time)
- • ln(10) ≈ 2.3026
- • ln(π) ≈ 1.1447
- • ln(√e) = 0.5