Logarithm Rules — Complete Log Properties Reference A complete reference of logarithm rules and properties. Covers the three main log laws (product, quotient, power), change of base, natural and common logs, and how to solve log equations. All rules assume positive arguments and a positive base ≠ 1.
Definition & Key Identities Log Definition
log a x = y ⟺ a y = x ( a > 0 , a ≠ 1 , x > 0 ) \log_a x = y \iff a^y = x \quad(a > 0,\ a \ne 1,\ x > 0) log a x = y ⟺ a y = x ( a > 0 , a = 1 , x > 0 ) Log of 1
log a 1 = 0 \log_a 1 = 0 log a 1 = 0 Log of the Base
log a a = 1 \log_a a = 1 log a a = 1 Log of a Power of the Base
log a a n = n \log_a a^n = n log a a n = n Base to a Log
a log a x = x a^{\log_a x} = x a l o g a x = x Common Log
log x = log 10 x \log x = \log_{10} x log x = log 10 x Natural Log
ln x = log e x \ln x = \log_e x ln x = log e x
Core Logarithm Rules Product Rule
log a ( M N ) = log a M + log a N \log_a(MN) = \log_a M + \log_a N log a ( M N ) = log a M + log a N Quotient Rule
log a ( M N ) = log a M − log a N \log_a\!\left(\frac{M}{N}\right) = \log_a M - \log_a N log a ( N M ) = log a M − log a N Power Rule
log a M p = p log a M \log_a M^p = p\,\log_a M log a M p = p log a M Reciprocal Rule
log a ( 1 M ) = − log a M \log_a\!\left(\frac{1}{M}\right) = -\log_a M log a ( M 1 ) = − log a M Change of Base & Inverse Relations Change of Base Formula
log a x = log b x log b a \log_a x = \frac{\log_b x}{\log_b a} log a x = log b a log b x To Natural Log
log a x = ln x ln a \log_a x = \frac{\ln x}{\ln a} log a x = ln a ln x To Common Log
log a x = log x log a \log_a x = \frac{\log x}{\log a} log a x = log a log x Log Reciprocal Base
log a b = 1 log b a \log_a b = \frac{1}{\log_b a} log a b = log b a 1 Solving Logarithmic Equations Equality of Logs
log a M = log a N ⟺ M = N \log_a M = \log_a N \iff M = N log a M = log a N ⟺ M = N Single Log Form
log a x = k ⟺ x = a k \log_a x = k \iff x = a^k log a x = k ⟺ x = a k Exponential to Log
a x = b ⟺ x = log a b a^x = b \iff x = \log_a b a x = b ⟺ x = log a b Combine to One Log
log a M + log a N = log a ( M N ) \log_a M + \log_a N = \log_a(MN) log a M + log a N = log a ( M N ) Expand a Log
log a M p = p log a M \log_a M^p = p \log_a M log a M p = p log a M Calculus Connections Derivative of ln x
d d x [ ln x ] = 1 x \frac{d}{dx}[\ln x] = \frac{1}{x} d x d [ ln x ] = x 1 Derivative of log_a x
d d x [ log a x ] = 1 x ln a \frac{d}{dx}[\log_a x] = \frac{1}{x \ln a} d x d [ log a x ] = x ln a 1 Integral of 1/x
∫ 1 x d x = ln ∣ x ∣ + C \int \frac{1}{x}\,dx = \ln|x| + C ∫ x 1 d x = ln ∣ x ∣ + C Maclaurin Series for ln(1+x)
ln ( 1 + x ) = x − x 2 2 + x 3 3 − ⋯ ( ∣ x ∣ < 1 ) \ln(1+x) = x - \tfrac{x^2}{2} + \tfrac{x^3}{3} - \cdots \quad(|x| < 1) ln ( 1 + x ) = x − 2 x 2 + 3 x 3 − ⋯ ( ∣ x ∣ < 1 )
