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?" The logarithm of x to base b is denoted as log_b(x).

2. How Does the Calculator Work?

The calculator uses the logarithm change of base formula:

Where:

• \( x \) — The value whose logarithm is calculated (must be positive)
• \( b \) — The logarithm base (must be positive and ≠ 1)
• \( \ln \) — Natural logarithm (base e)

Explanation: The 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 logarithms, set base to e (approximately 2.71828).

5. Frequently Asked Questions (FAQ)

Q1: Why can't the base be 1?
A: The function log₁(x) is undefined because 1 raised to any power is always 1, never x.

Q2: What's the difference between log and ln?
A: log typically means log₁₀ (common log), while ln means logₑ (natural log with base e ≈ 2.71828).

Q3: Can I calculate negative logarithms?
A: The logarithm function is only defined for positive real numbers (x > 0).

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

Q5: How is this related to exponential functions?
A: Logarithms and exponentials are inverse functions: if y = bˣ, then x = log_b(y).