ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4855 on D866: Standard real logarithm function

Reflection property of standard logarithm function

Formulation 0

Let $\log_a$ be the D866: Standard real logarithm function in base $a \in (0, \infty) \setminus \{ 1 \}$.
Let $x, y \in (0, \infty)$ each be a D5407: Positive real number.

Then \begin{equation} \log_a \frac{x}{y} = - \log_a \frac{y}{x} \end{equation}

Subresults

▶	R5662: Reflection property of standard logarithm function for a single positive real number

Proofs

Proof 0

Let $\log_a$ be the D866: Standard real logarithm function in base $a \in (0, \infty) \setminus \{ 1 \}$.
Let $x, y \in (0, \infty)$ each be a D5407: Positive real number.

Using R4826: Logarithm of a ratio, we have \begin{equation} \log_a \frac{x}{y} = \log_a x - \log_a y = - (\log_a y - \log_a x) = - \log_a \frac{y}{x} \end{equation} $\square$