# Sum of Logarithms/General Logarithm

## Theorem

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 $b$.

## Proof 1

 $\displaystyle \log_b x + \log_b y$ $=$ $\displaystyle \map {\log_b} {b^{\log_b x + \log_b y} }$ Definition of General Logarithm $\displaystyle$ $=$ $\displaystyle \map {\log_b} {\paren {b^{\log_b x} } \paren {b^{\log_b y} } }$ Exponent Combination Laws: Product of Powers $\displaystyle$ $=$ $\displaystyle \map {\log_b} {x y}$ Definition of General Logarithm

$\blacksquare$

## Proof 2

 $\displaystyle \log_b x + \log_b y$ $=$ $\displaystyle \frac {\ln x} {\ln b} + \frac {\ln y} {\ln b}$ Change of Base of Logarithm $\displaystyle$ $=$ $\displaystyle \frac {\ln x + \ln y} {\ln b}$ $\displaystyle$ $=$ $\displaystyle \frac {\map \ln {x y} } {\ln b}$ Sum of Logarithms: Proof for Natural Logarithm $\displaystyle$ $=$ $\displaystyle \map {\log_b} {x y}$ Change of Base of Logarithm

$\blacksquare$