Logarithm Calculator

Calculate logarithms of any base with step-by-step solutions and property explanations

Home Categories Math Logarithm Calculator

Expression:

log() Base for logb() = log()

Step-by-Step Solution

Logarithm Properties Reference

Property Formula

Common Logarithm Values

x log₁₀ ln log₂

Real-World Applications

Base Conversion Factors

1 / ln(10) ≈ 0.4343
ln(10) ≈ 2.3026
1 / ln(2) ≈ 1.4427
ln(2) ≈ 0.6931

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About Logarithm Calculator

What is a Logarithm?

A logarithm is the inverse operation of exponentiation. The logarithm of a number x to base b (written as log_b(x)) is the power to which b must be raised to equal x. If log_b(x) = y, then b^y = x.

How to Use This Calculator

  1. Select your calculation mode: Choose from Calculate Logarithm, Find Antilog, Find Base, or Change of Base
  2. Enter the required values: Input the number and base (where applicable)
  3. View instant results: See the result with step-by-step explanation
  4. Explore properties: Learn logarithm rules and conversions

Key Formulas

Logarithm Definition

  • If b^y = x, then log_b(x) = y
  • Example: log_10(100) = 2 because 10² = 100

Common Logarithm Bases

  • log (Common Log): Base 10, written as log₁₀(x) or log(x)
  • ln (Natural Log): Base e ≈ 2.71828, written as ln(x)
  • log₂ (Binary Log): Base 2, common in computer science

Change of Base Formula

log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b)

Logarithm Properties

Product Rule

log_b(xy) = log_b(x) + log_b(y)

Quotient Rule

log_b(x/y) = log_b(x) - log_b(y)

Power Rule

log_b(x^n) = n × log_b(x)

Change of Base

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

Special Values

  • log_b(1) = 0 for any base b > 0
  • log_b(b) = 1 for any base b > 0
  • log_b(b^n) = n for any base b > 0

Applications of Logarithms

Science & Engineering

  • pH scale (hydrogen ion concentration)
  • Decibel scale (sound intensity)
  • Richter scale (earthquake magnitude)
  • Radioactive decay calculations

Finance

  • Compound interest calculations
  • Growth rate analysis
  • Time value of money

Computer Science

  • Algorithm complexity (O(log n))
  • Binary search performance
  • Information theory (entropy)

Common Logarithm Values

x log₁₀(x) ln(x) log₂(x)
1 0 0 0
2 0.301 0.693 1
e 0.434 1 1.443
10 1 2.303 3.322
100 2 4.605 6.644
1000 3 6.908 9.966

Note: Logarithms are only defined for positive numbers when using real number bases. The base must be positive and not equal to 1.