Laws of Logarithms

Theorem

Let $x, y, b \in \R_{>0}$ be (strictly) positive real numbers.

Let $a \in \R$ be any real number such that $a > 0$ and $a \ne 1$.

Let $\log_a$ denote the logarithm to base $a$.

Then:

Change of Base of Logarithm

$\log_b x = \dfrac {\log_a x} {\log_a b}$

Sum of Logarithms

Let $x, y, b \in \R$ be strictly positive real numbers such that $b > 1$.

Then:

$\log_b x + \log_b y = \map {\log_b} {x y}$

where $\log_b$ denotes the logarithm to base $b$.

Logarithm of Power/Natural Logarithm

Let $x \in \R$ be a strictly positive real number.

Let $a \in \R$ be a real number such that $a > 1$.

Let $r \in \R$ be any real number.

Let $\ln x$ be the natural logarithm of $x$.

Then:

$\map \ln {x^r} = r \ln x$

Logarithm of Power/General Logarithm

Let $x \in \R$ be a strictly positive real number.

Let $a \in \R$ be a real number such that $a > 1$.

Let $r \in \R$ be any real number.

Let $\log_a x$ be the logarithm to the base $a$ of $x$.

Then:

$\map {\log_a} {x^r} = r \log_a x$

Difference of Logarithms

$\log_b x - \log_b y = \map {\log_b} {\dfrac x y}$

Logarithm of Reciprocal

$\map {\log_b} {\dfrac 1 x} = -\log_b x$

Examples

Arbitrary Example

$\ln \dfrac {A^p + B^q + C^r} {D^s E^t} = p \ln A + q \ln B + r \ln C - s \ln D - t \ln E$