Logarithm Calculator

Calculate Logarithm

Common Logarithm Values

Understanding Logarithms

What is a Logarithm?

A logarithm answers the question: "To what power must the base be raised to get x?"

Example: log₁₀(100) = 2 because 10² = 100

Types of Logarithms

• Common Log (log₁₀): Base 10, used in science and engineering
• Natural Log (ln): Base e ≈ 2.71828, used in calculus and growth models
• Binary Log (log₂): Base 2, used in computer science and information theory

Change of Base Formula

Convert a logarithm from one base to another:

Key Properties

• Product: log(xy) = log(x) + log(y)
• Quotient: log(x/y) = log(x) - log(y)
• Power: log(xⁿ) = n × log(x)
• Identity: log_b(b) = 1
• Zero: log_b(1) = 0

Domain Restrictions

• x must be positive (x > 0) — you cannot take log of zero or negative numbers
• Base must be positive and not equal to 1 (b > 0, b ≠ 1)

Real-World Applications

• Decibels: Sound intensity uses log₁₀ scale
• pH Scale: Acidity is -log₁₀ of hydrogen ion concentration
• Richter Scale: Earthquake magnitude
• Compound Interest: Time to double money
• Algorithm Complexity: O(log n) in computer science

Related Tools

• Permutation Calculator -- Calculate nPr permutations with step-by-step formula breakdown
• Speed Distance Time Calculator -- Calculate speed, distance, or time from any two known values with unit
• Decimal to Fraction Converter -- Convert decimals to fractions and vice versa
• Distance Formula Calculator -- Calculate distance between two points using the distance formula

Logarithm Calculator FAQ