Difference of Logarithms/Proof 3
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Theorem
- $\log_b x - \log_b y = \map {\log_b} {\dfrac x y}$
Proof
| \(\ds \map {\log_b} {\frac x y} + \log_b y\) | \(=\) | \(\ds \map {\log_b} {\frac x y \times y}\) | Sum of Logarithms | |||||||||||
| \(\ds \) | \(=\) | \(\ds \log_b x\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map {\log_b} {\frac x y}\) | \(=\) | \(\ds \log_b x - \log_b y\) | subtracting $\log_b y$ from both sides |
$\blacksquare$
Sources
- 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: Exercise $15$