Logarithm Calculator - Calculate Log, Ln & Any Base Logarithms with Step-by-Step Solutions

The base of the logarithm (must be positive and ≠ 1)

The number for which to find the logarithm (must be positive)

Common Logarithms:

  • • log₁₀(100) = 2
  • • ln(e) = 1
  • • log₂(8) = 3
  • • log₁₀(1) = 0 (any base)

Logarithm Result

Enter values and click Calculate to see results

Logarithm Properties & Rules

Product Rule

log_a(xy) = log_a(x) + log_a(y)

The log of a product equals the sum of the logs

Example:

log₁₀(100) = log₁₀(10 × 10)
= log₁₀(10) + log₁₀(10)
= 1 + 1 = 2

Quotient Rule

log_a(x/y) = log_a(x) - log_a(y)

The log of a quotient equals the difference of the logs

Example:

log₁₀(100/10) = log₁₀(100) - log₁₀(10)
= 2 - 1 = 1

Power Rule

log_a(x^n) = n × log_a(x)

The log of a power equals the exponent times the log

Example:

log₁₀(100²) = 2 × log₁₀(100)
= 2 × 2 = 4

Change of Base

log_a(x) = log_b(x) / log_b(a)

Convert logarithm to any base

Example:

log₂(8) = log₁₀(8) / log₁₀(2)
= 0.903 / 0.301 ≈ 3

Special Values

log_a(1) = 0

log_a(a) = 1

Important identities

Examples:

log₁₀(1) = 0
log₁₀(10) = 1
ln(1) = 0
ln(e) = 1

Inverse Property

a^(log_a(x)) = x

log_a(a^x) = x

Logarithm and exponent are inverses

Examples:

10^(log₁₀(5)) = 5
log₁₀(10³) = 3

About Logarithm Calculator

Our Logarithm Calculator is a comprehensive and free online tool designed to calculate common logarithms (log₁₀), natural logarithms (ln), binary logarithms (log₂), and logarithms with any custom base. Whether you're a student learning algebra, a scientist working with exponential data, or an engineer solving complex equations, our calculator provides instant, accurate results with detailed step-by-step explanations.

Understanding logarithms is essential for mathematics, science, engineering, and computer science. A logarithm answers the question: "To what power must we raise the base to get this number?" Our calculator not only computes logarithms but also helps you convert between exponential and logarithmic forms, making it an excellent learning tool.

Key Features

  • Common Logarithm (log₁₀): Calculate base-10 logarithms instantly
  • Natural Logarithm (ln): Compute base-e logarithms for calculus and science
  • Binary Logarithm (log₂): Essential for computer science and information theory
  • Custom Base Logarithms: Calculate logarithms with any positive base
  • Change of Base Formula: Automatically converts between different bases
  • Form Conversion: Convert between exponential (a^b = c) and logarithmic (log_a(c) = b) forms
  • Multiple Result Formats: View results in both logarithmic and exponential notation
  • Step-by-Step Solutions: Understand the calculation process with detailed explanations
  • Logarithm Rules Reference: Quick access to all important logarithm properties
  • 100% Free: No registration or payment required

Understanding Logarithms

What is a Logarithm?

A logarithm is the inverse operation of exponentiation. The logarithm base a of x, written as loga(x), answers the question: "What power must we raise a to in order to get x?"

Mathematically: If ay = x, then loga(x) = y

Types of Logarithms

  • Common Logarithm (log or log₁₀): Base 10 logarithm. Example: log₁₀(100) = 2
  • Natural Logarithm (ln or logₑ): Base e (≈2.71828) logarithm. Example: ln(e) = 1
  • Binary Logarithm (log₂): Base 2 logarithm. Example: log₂(8) = 3
  • Custom Base Logarithm (log_a): Any positive base. Example: log₅(125) = 3

Logarithm Properties and Rules

1. Product Rule

loga(xy) = loga(x) + loga(y)
The logarithm of a product equals the sum of the logarithms.
Example: log₁₀(100) = log₁₀(10 × 10) = log₁₀(10) + log₁₀(10) = 1 + 1 = 2

2. Quotient Rule

loga(x/y) = loga(x) - loga(y)
The logarithm of a quotient equals the difference of the logarithms.
Example: log₁₀(100/10) = log₁₀(100) - log₁₀(10) = 2 - 1 = 1

3. Power Rule

loga(xn) = n × loga(x)
The logarithm of a power equals the exponent times the logarithm.
Example: log₁₀(100²) = 2 × log₁₀(100) = 2 × 2 = 4

4. Change of Base Formula

loga(x) = logb(x) / logb(a)
Convert a logarithm to any base.
Example: log₂(8) = log₁₀(8) / log₁₀(2) = 0.903 / 0.301 ≈ 3

5. Special Values

loga(1) = 0 (any base raised to 0 equals 1)
loga(a) = 1 (any base raised to 1 equals itself)
Example: log₁₀(1) = 0, log₁₀(10) = 1

Practical Applications

  • Science: Measuring pH levels, earthquake magnitude (Richter scale), sound intensity (decibels)
  • Finance: Compound interest calculations, exponential growth models, investment analysis
  • Computer Science: Algorithm complexity analysis, binary search, data compression
  • Engineering: Signal processing, control systems, exponential decay models
  • Statistics: Log transformations, exponential distributions, survival analysis
  • Chemistry: Reaction kinetics, half-life calculations, chemical equilibrium
  • Biology: Population growth models, pharmacokinetics, enzyme kinetics
  • Physics: Entropy calculations, radioactive decay, wave behavior

Tips for Working with Logarithms

  1. Check Input Values: Remember that logarithms are only defined for positive numbers
  2. Use the Right Base: Choose log₁₀ for general calculations, ln for calculus and continuous growth
  3. Apply Rules Carefully: Use product, quotient, and power rules to simplify complex expressions
  4. Convert When Needed: Use change of base formula to work with unfamiliar bases
  5. Understand the Inverse: Remember that logarithms and exponentials are inverse operations
  6. Verify Results: Check your answer by converting back to exponential form

Common Logarithm Values Reference

Powers of 10:

log₁₀(1) = 0, log₁₀(10) = 1, log₁₀(100) = 2, log₁₀(1000) = 3, log₁₀(10000) = 4

Natural Logarithms:

ln(1) = 0, ln(e) ≈ 1, ln(e²) = 2, ln(10) ≈ 2.303

Powers of 2:

log₂(1) = 0, log₂(2) = 1, log₂(4) = 2, log₂(8) = 3, log₂(16) = 4, log₂(32) = 5

Why Use Our Logarithm Calculator?

Our Logarithm Calculator stands out for its versatility, accuracy, and educational value. Unlike basic calculators that only show decimal results, we provide comprehensive explanations in both logarithmic and exponential forms. The step-by-step solutions help you understand the mathematical process, making it an excellent tool for learning.

Whether you need to calculate a simple common logarithm or perform complex conversions between exponential and logarithmic forms, our calculator handles it all with ease. The intuitive interface works perfectly on desktop and mobile devices, ensuring you can solve logarithm problems wherever you are. Best of all, it's completely free with no registration required.