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📊 Logarithm Calculator

Calculate logₐ(x) with multiple base options

x must be positive

log10(100)
2
10^2 = 100
ln(100)
4.60517019
Natural Log
lg(100)
2
Common Log
log₂(100)
6.64385619
Binary Log
Change of Base Verification
log10(100) = ln(100) / ln(10)
= 4.60517019 / 2.30258509 = 2

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Logarithm Calculator Guide

The Logarithm Calculator supports computing logarithms with any base, with built-in shortcuts for common bases (2, e, 10). It simultaneously displays results for natural log (ln), common log (lg), and binary log (log₂) for easy comparison.

Logarithm is the inverse of exponentiation: if a^b = c, then logₐ(c) = b. Logarithms are extremely important in scientific computation — they convert multiplication to addition and exponentiation to multiplication, greatly simplifying complex calculations (this is how slide rules work). This tool is suitable for students learning logarithm concepts, engineers performing quick conversions, and researchers analyzing data.

Core Logarithm Formulas

【Definition】If a^b = x, then logₐ(x) = b

【Change of Base】logₐ(x) = ln(x)/ln(a) = lg(x)/lg(a)

【Basic Properties】
logₐ(M×N) = logₐM + logₐN    (Log of product = sum of logs)
logₐ(M/N) = logₐM - logₐN    (Log of quotient = difference of logs)
logₐMⁿ = n × logₐM           (Log of power = exponent × log)
logₐ1 = 0                     (Log of 1 is 0)
logₐa = 1                     (Log of base is 1)

【Common Values】
ln(10) ≈ 2.3026, lg(e) ≈ 0.4343
ln(2) ≈ 0.6931, log₂(10) ≈ 3.3219

Practical Examples

📋
【Example 1】Find lg(1000) Input x=100, base=10 Result: lg(1000) = 3 Verification: 10³ = 1000 ✓ 【Example 2】Find ln(e²) Input x=e²≈7.389, base=e Result: ln(7.389) ≈ 2 Verification: e² ≈ 7.389 ✓ 【Example 3】Find log₂(8) Input x=8, base=2 Result: log₂(8) = 3 Verification: 2³ = 8 ✓ 【Example 4】Custom base: log₃(81) Input x=81, custom base=3 Result: log₃(81) = 4 Verification: 3⁴ = 81 ✓

Important Notes

💡
- The argument x must be greater than 0 (negative numbers and 0 have no real logarithms) - The base a must be greater than 0 and not equal to 1 - Natural log ln uses base e≈2.71828, common log lg uses base 10 - The change of base formula is the universal tool for logarithm calculation, converting any base to a known base - In programming, Math.log() is the natural log, Math.log10() is the common log

Application Scenarios

  • - Chemistry pH: pH = -lg[H⁺], logarithmic measure of acidity
  • - Acoustics dB: dB = 10×lg(I/I₀), logarithmic scale of sound intensity
  • - Earthquake magnitude: Richter scale is a logarithmic function of seismic energy
  • - Information theory: Information entropy H = -Σp×log₂(p), measured in bits
  • - Finance: Doubling time t = ln(2)/ln(1+r)

Three Common Logarithms Compared

Natural logarithm (ln, base e): The most "natural" choice in mathematical analysis, because d/dx(ln x)=1/x has the simplest derivative form. e itself is defined as the unique positive number where ln(e)=1.

Common logarithm (lg, base 10): Based on our everyday counting system, commonly used in engineering for order-of-magnitude notation (e.g., kHz, MHz, GHz are powers of 10).

Binary logarithm (log₂, base 2): The foundation of computer science, directly related to bits. The log in algorithm time complexity O(log n) typically refers to log₂.

All three can be converted using the change of base formula. Mastering one allows you to derive results for the other two.

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