Business Mathematics and Basic Statistics · Ch 4 — Logarithms

Laws of Logarithms — Product, Quotient, and Power Rule

Because a logarithm is exactly an exponent, every law of logarithms below is really the corresponding law of INDICES, translated into logarithm language.

Product law

\log_a(MN) = \log_a M + \log_a N \qquad (M,N>0)

The log of a PRODUCT is the SUM of the logs — this mirrors the index product law ( $a^m\times a^n=a^{m+n}$ ), since multiplying two numbers corresponds to ADDING their exponents.

Quotient law

\log_a\left(\frac{M}{N}\right) = \log_a M - \log_a N \qquad (M,N>0)

The log of a QUOTIENT is the DIFFERENCE of the logs — mirroring the index quotient law.

Power law

\log_a(M^p) = p\,\log_a M \qquad (M>0)

The log of a number raised to a power $p$ is $p$ TIMES the log of that number — mirroring the index power-of-a-power law.

Note

The laws only combine logs of the SAME base

$\log_a M + \log_b N$ (different bases $a$ and $b$ ) cannot be combined into a single logarithm using these three laws — exactly as the index laws only combine powers of the same base. Always check that every logarithm in an expression shares the same base before applying the product, quotient, or power law.

A worked simplification pattern …

Definition 1Product Law of Logarithms

$\log_a(MN)=\log_a M+\log_a N$ — the log of a product is the sum of the logs (same ba …

Definition 2Quotient Law of Logarithms

$\log_a\left(\dfrac{M}{N}\right)=\log_a M-\log_a N$ — the log of a quotient is the difference of the logs (sa …

Definition 3Power Law of Logarithms

$\log_a(M^p)=p\log_a M$ — the log of a power is that power multiplied by the log of t …

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