Logarithm Calculator

Calculate logarithms with any base: log₁₀, ln (natural log), log₂, or custom base. Includes antilog calculator and logarithm rules reference.

Formula:log_b(x) = y means b^y = x

Result

log_10

2

Expressionlog_10(100)
Verification10^2 = 100

Logarithm Type

Calculate Logarithm

log10(
) = 2

Verification:

102.000000 = 100

Antilogarithm (Inverse)

Antilog is the inverse of log: if log_b(x) = y, then antilog_b(y) = x = b^y

10^
= 100

Graph of log₁₀(x)

Note: log(1) = 0 for all bases, log crosses x-axis at x = 1

Logarithm Rules

Product Rule

log(ab) = log(a) + log(b)

Quotient Rule

log(a/b) = log(a) - log(b)

Power Rule

log(aⁿ) = n · log(a)

Zero Rule

log(1) = 0

Identity Rule

log_b(b) = 1

Change of Base

log_b(x) = log(x)/log(b)

Common Logarithms

log₁₀ (Common)

log₁₀(1)0
log₁₀(10)1
log₁₀(100)2
log₁₀(1000)3
log₁₀(0.1)-1
log₁₀(0.01)-2

ln (Natural)

ln(1)0.0000
ln(e)1.0000
ln(e²)2.0000
ln(10)2.3026
ln(0.5)-0.6931
ln(2)0.6931

log₂ (Binary)

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

Important Constants

e (Euler's Number)

2.71828182845904...

Base of natural logarithms

log₁₀(e)

0.43429448190325...

Common log of e

ln(10)

2.30258509299404...

Natural log of 10

log₂(10)

3.32192809488736...

Binary log of 10

Result

log_10

2

Expressionlog_10(100)
Verification10^2 = 100

?How Do You Calculate Logarithms?

A logarithm answers 'what power gives this value?' If log_b(x) = y, then b^y = x. Common logs: log10(1000) = 3 (10^3 = 1000), ln(e) = 1 (e^1 = e). Key rules: log(ab) = log(a) + log(b), log(a/b) = log(a) - log(b), log(a^n) = n*log(a). Change of base: log_b(x) = log(x)/log(b).

What is a Logarithm?

A logarithm is the inverse operation of exponentiation. The logarithm base b of a number x (written log_b(x)) is the exponent to which b must be raised to produce x. Logarithms simplify multiplication to addition, making them historically crucial for calculations and still fundamental in mathematics, science, and engineering.

Key Facts About Logarithms

  • Logarithm: if log_b(x) = y, then b^y = x (inverse of exponentiation)
  • log (or log10): logarithm base 10. log10(100) = 2 because 10^2 = 100
  • ln: natural logarithm (base e where e is approximately 2.71828)
  • log2: logarithm base 2, used in computer science
  • Product rule: log(ab) = log(a) + log(b)
  • Quotient rule: log(a/b) = log(a) - log(b)
  • Power rule: log(a^n) = n*log(a)
  • Change of base: log_b(x) = log(x)/log(b) = ln(x)/ln(b)

Quick Answer

A logarithm answers 'what power gives this value?' If log_b(x) = y, then b^y = x. Common logs: log10(1000) = 3 (10^3 = 1000), ln(e) = 1 (e^1 = e). Key rules: log(ab) = log(a) + log(b), log(a/b) = log(a) - log(b), log(a^n) = n*log(a). Change of base: log_b(x) = log(x)/log(b).

Frequently Asked Questions

A logarithm answers the question: "To what power must the base be raised to get this number?" For example, log₂(8) = 3 because 2³ = 8. Logarithms are the inverse of exponential functions.
log (common logarithm) uses base 10 and is used in science and engineering. ln (natural logarithm) uses base e ≈ 2.718 and is used in calculus and natural growth. log₂ (binary logarithm) uses base 2 and is common in computer science.
Logarithms are undefined for zero and negative numbers. log(0) is undefined because no power of a positive base equals 0. log of negative numbers is undefined in real numbers (complex in advanced math). The base must be positive and not equal to 1.
Key rules: log(ab) = log(a) + log(b) (product), log(a/b) = log(a) - log(b) (quotient), log(aⁿ) = n·log(a) (power). Also: log(1) = 0, log(base) = 1, and change of base: log_b(x) = log(x)/log(b).
Use the change of base formula: log_b(x) = log(x) / log(b) = ln(x) / ln(b). Any common logarithm (base 10) or natural logarithm can be converted to any base using this formula.
e ≈ 2.71828 is the base of natural logarithms. It's a mathematical constant that appears in compound interest, population growth, and calculus. The function eˣ is unique because its derivative equals itself.

Last updated: 2025-01-15