㏒ Logarithm
Log Calculator
Calculate log base 10, natural log (ln), log base 2 and any custom base. Get antilog, step-by-step explanation and log rules reference.
Select base
Value (log₁₀ of)
Base
Quick examples
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—log₁₀ (x)
—ln (x)
—log₂ (x)
—Antilog (b^result)
—1 / result
—Change of base
Step-by-step solution
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Logarithm rules and identities
Logarithms satisfy several essential rules that allow simplification of complex expressions:
log(xy) = log(x) + log(y) — Product rule
log(x/y) = log(x) − log(y) — Quotient rule
log(xⁿ) = n × log(x) — Power rule
logₐ(x) = log(x)/log(a) — Change of base
log(x/y) = log(x) − log(y) — Quotient rule
log(xⁿ) = n × log(x) — Power rule
logₐ(x) = log(x)/log(a) — Change of base
log(1) = 0
Any base to the power 0 equals 1, so the log of 1 in any base is always 0.
log_b(b) = 1
Any base raised to the power 1 equals itself, so log of the base equals 1.
ln(e) = 1
Since e¹ = e, the natural log of e equals 1. This is a special case of the rule above.
log_b(x) = ln(x)/ln(b)
The change of base formula lets you calculate any log using a calculator that only has ln or log₁₀.
Frequently asked questions
What is a logarithm?
A logarithm answers: to what power must the base b be raised to produce x? log_b(x) = n means bⁿ = x. Logarithms are the inverse operation of exponentiation. For example, log₁₀(1000) = 3 because 10³ = 1000.
What is log base 10?
Log base 10 (common logarithm, written log or log₁₀) is the power to which 10 must be raised to equal a number. log(100) = 2 because 10² = 100. log(1000) = 3 because 10³ = 1000.
What is the natural logarithm (ln)?
The natural logarithm (ln) uses Euler's number e ≈ 2.71828 as its base. ln(x) = n means eⁿ = x. The natural log is extensively used in calculus, physics and finance because of e's special mathematical properties.
What is the change of base formula?
The change of base formula: log_b(x) = log(x)/log(b) = ln(x)/ln(b). This lets you calculate any logarithm using just common log (log₁₀) or natural log (ln) which most calculators provide.
What is antilog?
Antilog is the inverse of logarithm. Antilog_b(n) = bⁿ. For example, antilog₁₀(3) = 10³ = 1000. If log(x) = 2.5, then x = antilog(2.5) = 10^2.5 ≈ 316.23.
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