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 due to its natural properties in calculus and growth/decay problems.

2. How Does the Calculator Work?

The calculator uses the natural logarithm formula:

Where:

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

Explanation: The natural logarithm calculates the power to which e must be raised to obtain the number x.

3. Applications of Natural Logarithm

Details: Natural logarithms are used in compound interest calculations, radioactive decay, population growth models, thermodynamics, and solving time constants in electrical circuits.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

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

Q2: What is ln(1)?
A: ln(1) = 0, because e^0 = 1.

Q3: What is ln(e)?
A: ln(e) = 1, because e^1 = e.

Q4: Can ln(x) be negative?
A: Yes, when 0 < x < 1, ln(x) is negative. For x > 1, ln(x) is positive.

Q5: Why is ln(0) undefined?
A: There's no real number y such that e^y = 0, as e^y approaches 0 only as y approaches negative infinity.