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, particularly in growth/decay problems and when dealing with exponential relationships.

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 "e raised to what power equals x?"

3. Importance of Natural Logarithm

Details: Natural logarithms are fundamental in calculus, complex analysis, and many scientific applications including compound interest, population growth, radioactive decay, and psychophysics (Weber-Fechner law).

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 means base 10 (common logarithm), while ln means base e (natural logarithm).

Q2: Can I calculate ln(0) or ln(-x)?
A: No, ln(0) is undefined and ln of negative numbers is not a real number (though it's defined in complex analysis).

Q3: What are common applications of ln?
A: Solving time constants in circuits, half-life calculations, pH in chemistry, and many statistical models.

Q4: How is ln related to exponential functions?
A: The natural logarithm is the inverse function of the exponential function: e^ln(x) = x.

Q5: What's the derivative of ln(x)?
A: The derivative is 1/x, which makes it fundamental in calculus.