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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

Learn More About Logarithms

Khan Academy

Video lessons and practice on logarithms and their properties

Math is Fun

Clear introduction to logarithms with interactive examples

OpenStax Algebra

Free textbook chapter on logarithmic functions and applications

Frequently Asked Questions

What is a logarithm?
A logarithm answers: 'To what power must the base be raised to get this number?' If log_b(x) = y, then b^y = x. Example: log₁₀(100) = 2 because 10² = 100.
What is the difference between ln and log?
ln (natural log) uses base e ≈ 2.718, common in calculus and the natural sciences. log (common log) usually means base 10, used in engineering and pH. log₂ (binary log) uses base 2 and is central to computer science.
What is the change of base formula?
log_b(x) = log_c(x) / log_c(b), where c is any convenient base. Most calculators use c = e or c = 10. Example: log₃(27) = ln(27)/ln(3) = 3.
What are the key logarithm properties?
Product rule: log(xy) = log(x) + log(y). Quotient rule: log(x/y) = log(x) − log(y). Power rule: log(xⁿ) = n·log(x). Also log_b(b) = 1 and log_b(1) = 0.
Can you take the log of a negative number?
In the real numbers log(x) is undefined for x ≤ 0; log is only defined for positive arguments. Complex logarithms extend the definition but require imaginary numbers.
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