1. What is Natural Logarithm?

The natural logarithm (ln) is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828459. It's the inverse of the exponential function.

2. How Does the Calculator Work?

The calculator computes the natural logarithm using PHP's built-in log() function:

Where:

• \( x \) — Input value (must be positive)
• \( e \) — Euler's number (~2.71828)

Explanation: The natural logarithm answers the question "e to what power equals x?"

3. Applications of Natural Logarithm

Details: Natural logarithms are widely used in mathematics, physics, engineering, and finance. They appear in compound interest calculations, radioactive decay, population growth models, and many natural phenomena.

4. Using the Calculator

Tips: Enter any positive number to calculate its natural logarithm. The result is unitless. For values ≤0, the logarithm is undefined.

5. Frequently Asked Questions (FAQ)

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

Q2: Can I calculate ln of zero or negative numbers?
A: No, ln(x) is only defined for x > 0. The function approaches -∞ as x approaches 0 from the right.

Q3: What is ln(1)?
A: ln(1) = 0, because e⁰ = 1.

Q4: What is ln(e)?
A: ln(e) = 1, because e¹ = e.

Q5: How is natural log used in real life?
A: It's used in calculating compound interest, modeling population growth, analyzing algorithm complexity, and in many scientific measurements like pH and decibels.