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

 $\ds \log_b x + \log_b y$ $=$ $\ds \map {\log_b} {b^{\log_b x + \log_b y} }$ Definition of General Logarithm $\ds$ $=$ $\ds \map {\log_b} {\paren {b^{\log_b x} } \paren {b^{\log_b y} } }$ Product of Powers $\ds$ $=$ $\ds \map {\log_b} {x y}$ Definition of General Logarithm

$\blacksquare$

Proof 2

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

$\blacksquare$

Notes

If one presupposes Exponent Combination Laws, the proofs for logarithms to the general base can be used to directly prove the laws for $\log_e$.

Whether this would be circular is ultimately dependent on which definition of $e^x$ one chooses.