1. What is a Logarithm?

A logarithm is the inverse operation to exponentiation, answering the question: "To what power must the base be raised to produce a given number?" It's widely used in mathematics, science, and engineering.

2. How Does the Calculator Work?

The calculator uses the logarithm 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: The formula allows calculation of logarithms with any positive base (except 1) using natural logarithms.

3. Common Logarithm Bases

Details:

• Base 10 (Common Logarithm): log₁₀(x), used in many scientific applications
• Base e (Natural Logarithm): ln(x), used in calculus and advanced mathematics
• Base 2 (Binary Logarithm): log₂(x), used in computer science

4. Using the Calculator

Tips:

• Enter positive values for both x and base
• The base cannot be 1 (log₁(x) is undefined)
• For natural log, set base to e (approximately 2.71828)

5. Frequently Asked Questions (FAQ)

Q1: Why can't the base be 1?
A: The function 1^y always equals 1, so there's no solution to log₁(x) unless x=1, and even then it's not uniquely defined.

Q2: What's the difference between log and ln?
A: log typically means log₁₀ (common logarithm), while ln means log_e (natural logarithm).

Q3: Can I calculate negative logarithms?
A: The logarithm is only defined for positive real numbers in real number calculations.

Q4: What are logarithm identities?
A: Important identities include:

• log_b(xy) = log_b(x) + log_b(y)
• log_b(x/y) = log_b(x) - log_b(y)
• log_b(x^y) = y·log_b(x)

Q5: Where are logarithms used in real life?
A: Applications include pH scale (chemistry), Richter scale (earthquakes), decibel scale (sound), and algorithmic complexity (computer science).