1. What is the Log Function?

The logarithm function log_b(x) answers the question "To what power must the base b be raised, to produce the number x?" It's the inverse of exponentiation.

2. How Does the Calculator Work?

The calculator uses the logarithm change of base formula:

Where:

• \( x \) — The number you want to take the logarithm of (must be positive)
• \( b \) — The base of the logarithm (must be positive and not equal to 1)
• \( \ln \) — The natural logarithm (base e)

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

3. Importance of Logarithm Calculation

Details: Logarithms are fundamental in mathematics, science, and engineering. They're used in decibel scales, pH calculations, earthquake magnitude scales, and many algorithms in computer science.

4. Using the Calculator

Tips: Enter a positive number for x and a positive number (not equal to 1) for the base. The calculator will compute the logarithm of x with base b.

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, so there's no solution to 1^y = x when x ≠ 1.

Q2: What are common logarithm bases?
A: Common bases are 10 (common logarithm), e ≈ 2.718 (natural logarithm), and 2 (binary logarithm).

Q3: What's the relationship between logarithms and exponents?
A: They are inverse functions: if b^y = x, then log_b(x) = y.

Q4: Can I calculate negative logarithms?
A: No, logarithms are only defined for positive real numbers.

Q5: How is this different from natural logarithm?
A: Natural logarithm uses base e, while this calculator lets you specify any valid base.