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Logarithm Calculator

Log Calculator

Calculate logarithms with any base including natural logarithm (ln) and common logarithm (log₁₀). Perfect for students, engineers, and researchers.

100% FreeChange of Base FormulaNatural & Common Logs
Logarithm Calculator
Calculate logarithms with any base using the change of base formula

Enter the number (must be > 0)

Choose base or enter custom value

Common Logarithm Examples
Click on any example to automatically fill the calculator
Example

Common logarithm: log₁₀(100) = 2

Number: 100
Base: 10
Example

Natural logarithm: ln(100) ≈ 4.605

Number: 100
Base: e
Example

Binary logarithm: log₂(8) = 3

Number: 8
Base: 2
Example

Base-3 logarithm: log₃(27) = 3

Number: 27
Base: 3
What is a Logarithm?

A logarithm is the inverse operation of exponentiation. If b^y = a (where b > 0, b ≠ 1), then y = log_b(a), which reads "the logarithm of a with base b."

Key Concepts:

  • Base: The number being raised to a power
  • Antilogarithm: The result of the exponential operation
  • Logarithm: The exponent that produces the antilogarithm
  • Domain: Only positive numbers have real logarithms

Example: 2³ = 8, so log₂(8) = 3
Formula: by=a⇔y=log⁡b(a)b^y = a \Leftrightarrow y = \log_b(a)

Natural vs Common Logarithms

Natural Logarithm (ln)

Base e ≈ 2.71828. Used in calculus, exponential growth/decay, and natural sciences.

Common Logarithm (log₁₀)

Base 10. Used in engineering, scientific notation, and pH calculations.

Applications:

  • Natural Log: Population growth, radioactive decay, compound interest
  • Common Log: pH values, earthquake magnitudes, decibel scales
Applications of Logarithms

Mathematics

  • Solving exponential equations
  • Calculus operations (derivatives, integrals)
  • Complex number analysis

Science & Engineering

  • pH calculations in chemistry
  • Signal processing and decibels
  • Half-life calculations in physics

Finance

  • Compound interest calculations
  • Investment growth analysis
  • Risk assessment models
Logarithm Properties

Basic Properties

• log⁡b(1)=0\log_b(1) = 0
• log⁡b(b)=1\log_b(b) = 1
• log⁡b(0)\log_b(0) is undefined

Operational Properties

• log⁡b(xy)=log⁡b(x)+log⁡b(y)\log_b(xy) = \log_b(x) + \log_b(y)
• log⁡b(x/y)=log⁡b(x)−log⁡b(y)\log_b(x/y) = \log_b(x) - \log_b(y)
• log⁡b(xk)=k⋅log⁡b(x)\log_b(x^k) = k \cdot \log_b(x)

Change of Base

log⁡b(a)=ln⁡(a)ln⁡(b)=log⁡10(a)log⁡10(b)\log_b(a) = \frac{\ln(a)}{\ln(b)} = \frac{\log_{10}(a)}{\log_{10}(b)}
Why Use Our Log Calculator?

Accurate Calculations

Uses precise change of base formula

Step-by-Step Solutions

Learn the calculation process

Multiple Bases

Natural, common, and custom bases

Free & Educational

No registration, comprehensive learning

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Log Calculator - Natural Logarithm & Common Logarithm Calculator | MathIsimple