Editing List of logarithmic identities (section)

==== Logarithm of a power ====

To state the ''logarithm of a power'' law formally:

:<math>\forall b \in \mathbb{R}_+, b \neq 1, \forall x \in \mathbb{R}_+, \forall r \in \mathbb{R},  \log_b(x^r) = r\log_b(x)</math>

Derivation:

Let <math>b \in \mathbb{R}_+</math>, where <math>b \neq 1</math>, let <math>x\in \mathbb{R}_+</math>, and let <math>r \in \mathbb{R}</math>. For this derivation, we want to simplify the expression <math>\log_b(x^r)</math>. To do this, we begin with the simpler expression <math>\log_b(x)</math>. Since we will be using <math>\log_b(x)</math> often, we will define it as a new variable: Let <math>m = \log_b(x)</math>.

To more easily manipulate the expression, we rewrite it as an exponential. By definition, <math>m = \log_b(x)  \iff  b^m = x</math>, so we have

:<math>b^m = x</math>

Similar to the derivations above, we take advantage of another exponent law. In order to have <math>x^r</math> in our final expression, we raise both sides of the equality to the power of <math>r</math>:

:<math>
\begin{align}
(b^m)^r &= (x)^r \\
b^{mr} &= x^r
\end{align}
</math>

where we used the exponent law <math>(b^m)^r = b^{mr}</math>.

To recover the logarithms, we apply <math>\log_b</math> to both sides of the equality.

:<math>\log_b(b^{mr}) = \log_b(x^r)</math>

The left side of the equality can be simplified using a logarithm law, which states that <math>\log_b(b^{mr}) = mr</math>.

:<math>mr = \log_b(x^r)</math>

Substituting in the original value for <math>m</math>, rearranging, and simplifying gives

:<math>
\begin{align}
\left( \log_b(x) \right)r &= \log_b(x^r) \\
r\log_b(x) &= \log_b(x^r) \\
\log_b(x^r) &= r\log_b(x)
\end{align}
</math>

This completes the derivation.