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
Try These Examples
Common logarithm calculations
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
Logarithm Formula
Interactive formula breakdown
x-1 rounded cursor-pointer transition-all" data-var="log">log(x) = y, where x-1 rounded cursor-pointer transition-all" data-var="b">b is the base
log
Logarithm function
The logarithm operation - finds the exponent
Current value: 2
b
Base
The base of the logarithm
Current value: 10
x
Value
The number we are taking the logarithm of
Current value: 100
y
Result
The exponent: b^y = x
Current value: 2
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
Practice Logarithm Problems
Test your understanding with real problems
Practice with 8 problems to test your understanding.
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
How this works
Formulas follow standard definitions from the NIST Digital Library of Mathematical Functions and classical textbook derivations. Calculations run entirely in your browser. Where a closed-form solution exists, it is used; where an iterative or numerical method is required, the implementation is named on the page.
Sources
- [1]NIST Digital Library of Mathematical FunctionsAcademicdlmf.nist.govAccessed Apr 21, 2026
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Full-stack software engineer specializing in embedded systems, web architecture, and AI/ML. Founder of Practical Web Tools. Built the gesture-controlled drone IP acquired by KD Interactive (Aura Drone, sold on Amazon).
Result
log_10
2
Expressionlog_10(100)
Verification10^2 = 100