1. What is Natural Logarithm?

The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is Euler's number (approximately 2.71828). It's the inverse function of the exponential function e^x.

2. How Does the Calculator Work?

The calculator uses the natural logarithm function:

Where:

• \( x \) — Input value (must be positive)

Explanation: The natural logarithm calculates the power to which e must be raised to equal the input value x.

3. Applications of Natural Logarithm

Details: Natural logarithms are widely used in mathematics, physics, engineering, and finance. They appear in compound interest calculations, population growth models, radioactive decay, and many natural phenomena.

4. Using the Calculator

Tips: Enter a positive value for x. The calculator will return the natural logarithm of that value. The result is unitless.

5. Frequently Asked Questions (FAQ)

Q1: Why must the input value be positive?
A: The natural logarithm function is only defined for positive real numbers. Logarithms of zero or negative numbers are undefined in the real number system.

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

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

Q4: How is natural logarithm different from common logarithm?
A: Natural logarithm uses base e (≈2.718), while common logarithm uses base 10. They're related by the formula: ln(x) = log(x) / log(e).

Q5: Can I calculate natural logarithm of very small numbers?
A: Yes, but as x approaches 0, ln(x) approaches negative infinity. Very small positive values will yield large negative results.