1Cademy - Summary of the Properties of Logarithms

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Power Property of Logarithms

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Summary of the Properties of Logarithms

The properties of logarithms provide a complete set of algebraic rules for evaluating, expanding, and condensing logarithmic expressions. For any valid base $a$ ( $a > 0$ and $a eq 1$ ), positive real numbers $M$ and $N$ , and any real number $p$ , the core properties are summarized as follows:

Logarithm of $1$ Property: $\log_a 1 = 0$ (Natural logarithm: $\ln 1 = 0$ )
Logarithm of the Base Property: $\log_a a = 1$ (Natural logarithm: $\ln e = 1$ )
Inverse Properties: $a^{\log_a x} = x$ and $\log_a a^x = x$ (Natural logarithm: $e^{\ln x} = x$ and $\ln e^x = x$ )
Product Property: $\log_a(M \cdot N) = \log_a M + \log_a N$ (Natural logarithm: $\ln(M \cdot N) = \ln M + \ln N$ )
Quotient Property: $\log_a \frac{M}{N} = \log_a M - \log_a N$ (Natural logarithm: $\ln \frac{M}{N} = \ln M - \ln N$ )
Power Property: $\log_a M^p = p \log_a M$ (Natural logarithm: $\ln M^p = p \ln M$ )