Laws of Logarithms

From ProofWiki
Jump to: navigation, search

Theorem

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

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


Then:

Change of Base of Logarithm

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


Sum of Logarithms

Theorem

Natural Logarithm

Let $x, y \in \R$ be strictly positive real numbers.


Then:

$\ln x + \ln y = \ln \left({x y}\right)$

where $\ln$ denotes the natural logarithm.


General Logarithm

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


Then:

$\log_b x + \log_b y = \log_b \left({x y}\right)$

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


Sources

Logarithms of Powers

$\log_a \left({x^r}\right) = r \log_a x$


Difference of Logarithms

$\log_b x - \log_b y = \log_b \left({\dfrac x y}\right)$


Logarithm of Reciprocal

$\log_b \left({\dfrac 1 x}\right) = - \log_b x$


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