Logarithm of Power

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Theorem

Natural Logarithms

Let $x \in \R$ be a strictly positive real number.

Let $a \in \R$ be a real number such that $a > 1$.

Let $r \in \R$ be any real number.

Let $\ln x$ be the natural logarithm of $x$.


Then:

$\ln \left({x^r}\right) = r \ln x$


General Logarithms

Let $x \in \R$ be a strictly positive real number.

Let $a \in \R$ be a real number such that $a > 1$.

Let $r \in \R$ be any real number.

Let $\log_a x$ be the logarithm to the base $a$ of $x$.


Then:

$\log_a \left({x^r}\right) = r \log_a x$


Sources