1. What Is Natural Logarithm?

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 because of its natural properties in calculus and growth/decay problems.

2. How Natural Log Calculation Works

The calculator uses the mathematical 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?" It's the inverse of the exponential function.

3. Applications of Natural Logarithm

Details: Natural logs are used in compound interest, population growth, radioactive decay, pH calculations, and solving time constants in physics and engineering.

4. Using the Calculator

Tips: Enter any positive real number. The result will be the power to which e must be raised to obtain your input value.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between ln and log?
A: ln is base e, while log typically refers to base 10 (unless specified otherwise in mathematics).

Q2: Can ln be calculated for negative numbers?
A: No, ln(x) is only defined for x > 0. For complex numbers, the definition extends but isn't covered here.

Q3: What's ln(1)?
A: ln(1) = 0, because e^0 = 1.

Q4: What's ln(e)?
A: ln(e) = 1, because e^1 = e.

Q5: Why is e the "natural" base?
A: Because functions with base e have simpler derivatives and integrals, making them naturally suited for calculus.