Sum of Logarithms/General Logarithm

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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

$\blacksquare$


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.


Sources