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, particularly in growth/decay problems and calculus.

2. How Does the Calculator Work?

The calculator computes the natural logarithm using the formula:

Where:

• \( x \) — Input value (must be positive)
• \( \ln \) — Natural logarithm function (base e)

Explanation: The natural logarithm of a number is the power to which e must be raised to obtain that number.

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 (ln(0) is undefined and ln(x) for x<0 is not a real number).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between ln and log?
A: ln is base e (≈2.718), while log typically refers to base 10. ln is the inverse of the exponential function e^x.

Q2: Why is ln(1) equal to 0?
A: Because e^0 = 1. Any logarithm of 1 is 0, regardless of base.

Q3: What does it mean if ln(x) is negative?
A: It means x is between 0 and 1. ln(x) approaches -∞ as x approaches 0.

Q4: Can I calculate ln of very large numbers?
A: Yes, but extremely large numbers may exceed computational limits. The function grows slowly (ln(1,000,000) ≈ 13.8155.

Q5: How is ln related to derivatives and integrals?
A: The derivative of ln(x) is 1/x, making it fundamental in calculus. It's also used in integration of 1/x dx = ln|x| + C.