1. What is a Logarithm?

The logarithm (log) is the inverse operation to exponentiation. It answers the question: "To what power must the base be raised to produce a given number?" The expression log_b(x) = y means that b^y = x.

2. How the Calculator Works

The calculator uses the 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: This formula allows calculation of logarithms with any base using the natural logarithm function available on most calculators.

3. Common Logarithm Bases

Common Bases:

• Base 10 (Common Logarithm): log₁₀(x), often written as log(x)
• Base e (Natural Logarithm): log_e(x), written as ln(x)
• Base 2 (Binary Logarithm): log₂(x), used in computer science

4. Using the Calculator

Tips:

• Enter a positive number for x (cannot be zero or negative)
• Enter a positive base not equal to 1
• For natural log, set base to 2.71828 (e)
• For common log, set base to 10

5. Frequently Asked Questions (FAQ)

Q1: Why can't the base be 1?
A: The function 1^y always equals 1, so log₁(x) is undefined for x ≠ 1 and indeterminate for x = 1.

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

Q3: How do I calculate logarithms without a calculator?
A: Before calculators, people used logarithm tables or slide rules. Some values can be estimated using known log values and logarithm properties.

Q4: What are logarithm properties?
A: Key properties 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?
A: Logarithms are used in many fields including mathematics, physics, chemistry, computer science, engineering, and finance for dealing with exponential relationships.