In this Wolfram webpage, under the section "For compositions with elementary functions" they give the following identities:
Power functions:
Log+power functions:
However, in those equations, the variables $m$ and $c$ in the right hand side seem to be undefined. Are they standard variables? What do they represent?
$f$ is an elementary function there, so analytic. Although it is not stated explicitly (unless I overlooked it), it is assumed that $f$ doesn't vanish identically. Thus $f$ has a power series expansion
$$f(z) = \sum_{n = 0} a_n \cdot (z-z_0)^n$$
valid in a neighbourhood of $z_0$, where not all coefficients $a_n$ vanish.
Possibly, they allow for $f$ to have a pole at $z_0$, then $f$ has a Laurent series expansion
$$f(z) = \sum_{n\in \mathbb{Z}} a_n\cdot (z-z_0)^n$$
about $z_0$ with finite principal part - that is, only finitely many $a_n$ with a negative index $n$ may be nonzero.
$m$ is the smallest index for which the coefficient is nonzero, and $c$ is the corresponding coefficient,
$$m = \min \{ n : a_n \neq 0\},\quad\text{and}\quad c = a_m.$$
Thus we can write
$$f(z) = c(z-z_0)^m\cdot g(z)\tag{1}$$
with an analytic function
$$g(z) = \sum_{n = 0}^\infty \frac{a_{n+m}}{a_m}(z - z_0)^n$$
satisfying $g(z_0) = 1$. The principal branches of powers and logarithms have simple expansions in neighbourhoods of $1$, and so the decomposition $(1)$ together with the known behaviour of powers and logarithms for products yields the given series expansions, where the last series is the expansion of the principal branch of the $b$-th power resp. logarithm of
$$c^{-r}\biggl(\frac{f(z)}{(z-z_0)^m}\biggr)^r = g(z)^r = 1 + \bigl( g(z)^r-1\bigr).$$