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 $b$.

$\ds \log_b x + \log_b y$

$=$

$\ds \map {\log_b} {b^{\log_b x + \log_b y} }$

Definition of General Logarithm

$\ds $

$=$

$\ds \map {\log_b} {\paren {b^{\log_b x} } \paren {b^{\log_b y} } }$

$\ds $

$=$

$\ds \map {\log_b} {x y}$

Definition of General Logarithm

$\blacksquare$

$\ds \log_b x + \log_b y$

$=$

$\ds \frac {\ln x} {\ln b} + \frac {\ln y} {\ln b}$

$\ds $

$=$

$\ds \frac {\ln x + \ln y} {\ln b}$

$\ds $

$=$

$\ds \frac {\map \ln {x y} } {\ln b}$

$\ds $

$=$

$\ds \map {\log_b} {x y}$

$\blacksquare$

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.

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Laws of Logarithms: $7.10$
1972: Murray R. Spiegel and R.W. Boxer: Theory and Problems of Statistics (SI ed.) ... (previous) ... (next): Chapter $1$: Computations Using Logarithms
1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms: $(11)$