1. What is a Logarithm?

A logarithm answers the question: "To what power must the base be raised to get this number?" 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 (unitless)
• \( x \) — The number you want to find the logarithm of (unitless, x > 0)
• \( b \) — The logarithm base (unitless, b > 0, b ≠ 1)
• \( \ln \) — Natural logarithm (base e)

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

3. Common Logarithm Bases

Common Bases:

• Base 10 (common logarithm): log₁₀(x)
• Base e (natural logarithm): ln(x)
• Base 2 (binary logarithm): log₂(x)

4. Using the Calculator

Tips: Enter positive values for both x and b. The base cannot be 1. For natural log, set base to 2.71828 (e).

5. Frequently Asked Questions (FAQ)

Q1: Why can't the base be 1?
A: The function f(x) = 1^x is constant (always equals 1), so it's not a valid logarithmic base.

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

Q3: Can I calculate negative logarithms?
A: No, logarithms are only defined for positive real numbers.

Q4: What are logarithms used for?
A: Logarithms are used in many fields including mathematics, physics, chemistry, computer science, and engineering.

Q5: How do I calculate antilogarithms?
A: The antilogarithm is simply the base raised to the logarithm value: antilog_b(y) = b^y.