natural logarithm

The general formula for the derivative of \(f(x)^g(x)\), with \(f\) and \(g\) differentiable, is as follows:

\[
D[f(x)]^{g(x)} = f(x)^{g(x)} \left[ g'(x) \ln f(x) + g(x) \frac{f'(x)}{f(x)} \right ]
\]

A logarithmic function is defined as a function of the form:

\[ y = \log{_a}x \quad \text{with} \quad a \gt 0 \quad a \neq 1 \quad \forall x \in \mathbb{R}^+\]