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Common logarithm calculations
Logarithm Type
Calculate Logarithm
Verification:
102.000000 = 100
Antilogarithm (Inverse)
Antilog is the inverse of log: if log_b(x) = y, then antilog_b(y) = x = b^y
Logarithm Formula
Interactive formula breakdown
The logarithm operation - finds the exponent
The base of the logarithm
The number we are taking the logarithm of
The exponent: b^y = x
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)
ln (Natural)
log₂ (Binary)
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
?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
What is a logarithm?
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.
What is the difference between log, ln, and log₂?
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.
When is the logarithm undefined?
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.
What are the main logarithm rules?
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).
How do I calculate log with any base?
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.
What is e (Euler's number)?
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: 2026-01-15
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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