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 base b is written as log_b(x) and is defined by:

2. How the Logarithm Calculation Works

This calculator uses the change of base formula:

Where:

• \( \ln(x) \) — Natural logarithm (base e) of x
• \( \ln(b) \) — Natural logarithm of the base
• The result is unitless

Explanation: The formula converts any logarithm to a ratio of natural logarithms, which can be easily calculated.

3. Practical Applications

Details: Logarithms are used in many fields including:

• Mathematics (solving exponential equations)
• Science (measuring earthquake intensity, sound levels)
• Computer science (algorithm complexity)
• Finance (compound interest calculations)

4. Using the Calculator

Tips:

• Enter positive values for both x and base
• The base cannot be 1 (logarithm undefined)
• For natural log (ln), use base e ≈ 2.71828
• For common log (base 10), use base 10

5. Frequently Asked Questions (FAQ)

Q1: What is the natural logarithm?
A: The natural logarithm (ln) has base e (Euler's number ≈ 2.71828). It's common in mathematics and physics.

Q2: What is the common logarithm?
A: The common logarithm has base 10, often written as log(x) without a base. Used in engineering and pH calculations.

Q3: Can the base be less than 1?
A: Yes, but it must be positive and not equal to 1. The function behaves differently for 0 < b < 1 versus b > 1.

Q4: Why are logarithms unitless?
A: Logarithms measure the exponent needed to produce a number, and exponents are always unitless quantities.

Q5: What about logarithms of negative numbers?
A: Real-valued logarithms are only defined for positive numbers. Complex logarithms can handle negative numbers.