Laws of Logarithmic Operations

Three fundamental laws convert the arithmetic of numbers into the arithmetic of their logarithms — turning multiplication into addition, division into subtraction, and powers into multiplication. This conversion is precisely what once made logarithms indispensable for heavy computation, and it is the reason they still simplify commercial calculations (§7). Throughout, $M$ and $N$ are positive numbers and $a$ is a valid base.

1. Product Law — the logarithm of a product is the sum of the logarithms:

Proof. Let $\log_{a}M = x$ and $\log_{a}N = y$, so $M = a^{x}$ and $N = a^{y}$. Then $M \times N = a^{x} \times a^{y} = a^{x+y}$, which in logarithmic form is $\log_{a}(MN) = x + y = \log_{a}M + \log_{a}N$. The law extends to any number of factors.

2. Quotient Law — the logarithm of a quotient is the difference of the logarithms:

Proof. With $M = a^{x}$ and $N = a^{y}$ as above, $\dfrac{M}{N} = \dfrac{a^{x}}{a^{y}} = a^{x-y}$, so $\log_{a}(M/N) = x - y = \log_{a}M - \log_{a}N$.

3. Power Law — the logarithm of a power is the exponent times the logarithm:

Proof. With $M = a^{x}$, we have $M^{p} = (a^{x})^{p} = a^{px}$, so $\log_{a}(M^{p}) = px = p\log_{a}M$. Since a root is a fractional power, this law also handles roots: $\log_{a}\sqrt[n]{M} = \log_{a}(M^{1/n}) = \dfrac{1}{n}\log_{a}M$.

Change of base. A logarithm to one base can be re-expressed in another base $b$:

This is what allows any logarithm to be evaluated using base-10 tables. A useful special case, obtained by setting $b = N = a$ appropriately, is the reciprocal relation …