Difference of Logarithms
Jump to navigation
Jump to search
Theorem
Let $x, y, b \in \R$ be strictly positive real numbers such that $b > 1$.
Then:
- $\log_b x - \log_b y = \map {\log_b} {\dfrac x y}$
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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\log_b} {\frac {\paren {b^{\log_b x} } } {\paren {b^{\log_b y} } } }\) | Quotient of Powers | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\log_b} {\frac x y}\) | Definition of General Logarithm |
$\blacksquare$
Proof 2
\(\ds \log_b x - \log_b y\) | \(=\) | \(\ds \frac {\log_e x} {\log_e b} - \frac {\log_e y} {\log_e b}\) | 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 |
$\blacksquare$
Proof 3
\(\ds \map {\log_b} {\frac x y} + \log_b y\) | \(=\) | \(\ds \map {\log_b} {\frac x y \times y}\) | 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
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Laws of Logarithms: $7.11$
- 1972: Murray R. Spiegel and R.W. Boxer: Theory and Problems of Statistics (SI ed.) ... (previous) ... (next): Chapter $1$: Computations Using Logarithms