An equation that occurs in ecology In the field of ecology, a well known relation between the number of species and the size of an island can be approximated by a power function of the form:
$S = c  A^z$


*

*$S$ = number of species

*$c$ = a fitted constant

*$A$ = area of the island

*$z$ = a constant equal to $\log_{10}(S) / \log_{10}(A)$


Variable $z$ is, in fact, the slope of the linear relation between $\log_{10}(S)$ and $\log_{10}(A)$.  I’ve never encountered a multivariate equation where one of the independent variables consists of the slope of a linear relation where a non-linear transform has been applied to the major independent and dependent variable.  
I’m curious if the previously mentioned equation belongs to a more general class of equations?  Is there a special name for these types of equations?
 A: In engineering fields we simply call this a power law relationship. Power law representations are almost always empirical.
Your equation isn't really multivariate. Its better to think of it as a parameterized single variable relation, where $z$ is computed from a body of data outside the scope of the studied region; alternatively, $A$ is fixed for the region in question and $z$ is estimated from experiments.
Regardless, the parameters are fixed in any given context and a single variable is fit to data.
Power law relationships often do not have a causal mechanism such that they take that form, and as such only apply over a specific domain. These models are used widely in engineering fields, and substantial effort must usually be done to justify their prognostic abilities.
A: Are you sure you're not missing something? Because from what you said it can be derived that 
$S=cA^z$
but on the other hand you say we have $z=\frac{\lg S}{\lg A}=\log _A \left( S \right) $ 
so we have $S=c\cdot S$.
A: Please look at the Wikipedia entry on allometry. Variable $z$ is estimated from the data. 
