1. What is Natural Logarithm?

The natural logarithm (ln) is the logarithm to the base e (Euler's number, approximately 2.71828). It's defined as the area under the curve y = 1/t from 1 to x. The natural logarithm is fundamental in mathematics, especially in calculus and complex analysis.

2. How is Natural Log Calculated?

The natural logarithm is defined by the integral:

Where:

• \( x \) — Positive real number (input value)
• \( t \) — Variable of integration
• \( \ln(x) \) — Natural logarithm of x (output)

Explanation: The natural logarithm measures the time needed to reach a certain level of continuous growth or the area under a hyperbola.

3. Properties of Natural Logarithm

Key Properties:

• \(\ln(1) = 0\)
• \(\ln(e) = 1\) where e ≈ 2.71828
• \(\ln(ab) = \ln(a) + \ln(b)\)
• \(\ln(a^b) = b\ln(a)\)
• \(\ln\) is undefined for non-positive numbers

4. Using the Calculator

Tips: Enter any positive real number to calculate its natural logarithm. The input must be greater than 0.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between log and ln?
A: "log" typically means base 10 logarithm, while "ln" is the natural logarithm with base e.

Q2: Can I calculate ln(0) or ln(-1)?
A: No, the natural logarithm is only defined for positive real numbers.

Q3: Why is e the base for natural logarithms?
A: The base e arises naturally in calculus and has unique properties in differentiation and integration.

Q4: How is ln(x) calculated numerically?
A: Computers use approximation methods like Taylor series or arithmetic-geometric mean iterations.

Q5: What are practical applications of ln?
A: Used in compound interest, population growth, radioactive decay, and many scientific formulas.