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
\(\ds \log_b x + \log_b y\) | \(=\) | \(\ds \map {\log_b} {b^{\log_b x + \log_b y} }\) | 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.
Sources
- 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: $(11)$