Difference of Logarithms/Proof 2
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Theorem
- $\log_b x - \log_b y = \map {\log_b} {\dfrac x y}$
Proof
| \(\ds \log_b x - \log_b y\) | \(=\) | \(\ds \frac {\log_e x} {\log_e b} - \frac {\log_e y} {\log_e b}\) | Change of Base of Logarithm | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\log_e x - \log_e y} {\log_e b}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\log_e \left({\frac x y}\right)} {\log_e b}\) | Difference of Logarithms: Proof for Natural Logarithm | |||||||||||
| \(\ds \) | \(=\) | \(\ds \log_b \left({\frac x y}\right)\) | Change of Base of Logarithm |
$\blacksquare$