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

Result R5662 on D866: Standard real logarithm function
Subresult of R4855: Reflection property of standard logarithm function

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

Formulation 0

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

Then \begin{equation} \log_a x = - \log_a \frac{1}{x} \end{equation}

Proofs

Proof 0

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

This result is a special case of R4855: Reflection property of standard logarithm function with $y = 1$. $\square$