1. What is Natural Logarithm (ln)?

The natural logarithm (ln) is the logarithm to the base e (Euler's number, approximately 2.71828). It's widely used in mathematics, physics, and engineering to solve problems involving growth and decay.

2. How Does the Calculator Work?

The calculator computes the natural logarithm using the PHP log() function:

Where:

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

Explanation: The natural logarithm answers the question "To what power must e be raised to get x?"

3. Importance of Natural Logarithm

Details: Natural logarithms are fundamental in calculus, complex analysis, and modeling exponential growth/decay processes in science and finance.

4. Using the Calculator

Tips: Enter any positive number (x > 0) to calculate its natural logarithm. The result is unitless.

5. Frequently Asked Questions (FAQ)

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

Q2: Why can't ln(x) be calculated for x ≤ 0?
A: The natural logarithm function is only defined for positive real numbers.

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 ln(x) related to exponential functions?
A: The natural logarithm is the inverse function of the exponential function: e^ln(x) = x.