1. What is the 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 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)

Explanation: The natural logarithm is the inverse of the exponential function \( e^x \).

3. Importance of Natural Logarithm

Details: Natural logarithms are essential in calculus, complex analysis, and modeling continuous growth processes like population growth, radioactive decay, and compound interest.

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 is base e (natural log), while log typically refers to base 10 (common log) unless specified otherwise.

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: The result can be negative when 0 < x < 1, but ln(x) is undefined for x ≤ 0.

Q5: Where is ln(x) used in real life?
A: In finance for continuous compounding, in physics for radioactive decay, and in statistics for log-normal distributions.