Relations of the form f(x) = k⋅x^{h} are called
power law relations. 
Science is filled with power laws. For instance, 
Hooke's law for springs: F(x) = k⋅x 
Newton's law of gravitation: F(r) = GMm⋅r^{2} 
the allometry of animal metabolic rates: metabolic rate =
k⋅(weight)^{3/4}. It turns out this is a bit of a
surprise. 

By themselves, power laws do not imply fractal structure. 
For example, the Stellpflug formula relating weight and radius of
pumpkins is 
weight = k⋅radius^{2.78} 
yet no one would say a pumpkin is a fractal. 
A pumpkin is a
roughly spherical shell enclosing an empty cavity; a pumpkin certainly isn't made
up of smaller pumpkins. 
Rather, this is the scaling relation between the thickness
of the shell and the size of the pumpkin. 
Here are two examples where a power law does give a dimension. 

Here is another example, illustrating how power law plots
can reveal fractal patterns present when random arrangements obscure
strict geometric hierarchies. 