1. What is Natural Logarithm?

The natural logarithm (ln) is the logarithm to the base e (Euler's number, approximately 2.71828). It's defined as the area under the curve y = 1/t from 1 to x, or equivalently as the inverse of the exponential function.

2. How Does the Calculator Work?

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

Where:

• \( x \) — Positive real number (x > 0)
• \( \ln(x) \) — Natural logarithm of x (dimensionless)

Explanation: The natural logarithm grows slowly as x increases and approaches negative infinity as x approaches 0.

3. Importance of Natural Logarithm

Details: Natural logarithms are fundamental in mathematics, physics, and engineering. They appear in growth/decay processes, entropy calculations, complex analysis, and many natural phenomena.

4. Using the Calculator

Tips: Enter any positive number (x > 0). The calculator will return the natural logarithm of that number. For numbers ≤ 0, the result is undefined.

5. Frequently Asked Questions (FAQ)

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

Q2: What is ln(1)?
A: ln(1) = 0, since e^0 = 1. This follows from the integral definition as the area from 1 to 1 is zero.

Q3: Why is ln(0) undefined?
A: The integral of 1/t from 1 to 0 diverges to negative infinity, and e^x never equals 0 for any finite x.

Q4: What are some important properties of ln?
A: Key properties include: ln(ab) = ln(a) + ln(b), ln(a^b) = b·ln(a), and ln(e) = 1.

Q5: Where is ln used in real-world applications?
A: Applications include compound interest, population growth, radioactive decay, pH calculations, and signal processing.