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

Result R4831 on D866: Standard real logarithm function

Homomorphism 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 $x_1, \ldots, x_N \in (0, \infty)$ each be a D5407: Positive real number.

Then \begin{equation} \log_a \left( \prod_{n = 1}^N x_n \right) = \sum_{n = 1}^N \log_a x_n \end{equation}

Formulation 1

Let $\log_a$ be the D866: Standard real logarithm function in base $a \in (0, \infty) \setminus \{ 1 \}$.
Let $x_1, \ldots, x_N \in (0, \infty)$ each be a D5407: Positive real number.

Then \begin{equation} \log_a(x_1 x_2 \cdots x_N) = \log_a(x_1) + \log_a(x_2) + \cdots + \log_a(x_N) \end{equation}

Proofs

Proof 0

Let $\log_a$ be the D866: Standard real logarithm function in base $a \in (0, \infty) \setminus \{ 1 \}$.
Let $x_1, \ldots, x_N \in (0, \infty)$ each be a D5407: Positive real number.

Using result R3498: Homomorphism property of the standard natural logarithm, we have \begin{equation} \log_a \left( \prod_{n = 1}^N x_n \right) = \frac{\log \left( \prod_{n = 1}^N x_n \right)}{\log a} = \frac{\sum_{n = 1}^N \log x_n}{\log a} = \sum_{n = 1}^N \frac{\log x_n}{\log a} = \sum_{n = 1}^N \log_a x_n \end{equation} $\square$