1. What is a Logarithm?

A logarithm is the inverse operation to exponentiation, just as division is the inverse of multiplication. The logarithm of a number x with respect to base b is the exponent to which b must be raised to yield x.

2. How Does the Calculator Work?

The calculator uses the change of base formula:

Where:

• \( \log_b(x) \) — Logarithm of x with base b
• \( \ln(x) \) — Natural logarithm (base e) of x
• \( \ln(b) \) — Natural logarithm of the base

Explanation: This formula allows calculation of logarithms with any base using natural logarithms.

3. Common Logarithm Bases

Details:

• Base 10 (Common Logarithm): log₁₀(x), often written as log(x)
• Base e (Natural Logarithm): log_e(x), written as ln(x)
• Base 2 (Binary Logarithm): log₂(x), important in computer science

4. Using the Calculator

Tips: Enter positive numbers for both x and the base. The result is unitless. For natural logarithms, set base to 2.71828 (e).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between log and ln?
A: log typically means log₁₀ (common logarithm) while ln means log_e (natural logarithm).

Q2: Can I calculate logarithms with base 1?
A: No, base must be a positive number different from 1.

Q3: What is log_b(1) for any base?
A: Always 0, because b⁰ = 1 for any valid base b.

Q4: What is log_b(b)?
A: Always 1, because b¹ = b.

Q5: How are logarithms used in real life?
A: Used in many fields including science (pH scale), finance (compound interest), and computer science (algorithm complexity).