I would like to know what are the especifications of a functional equation that give us a power function as a solution.
For example, if $f:\Bbb R \to \Bbb R$ is continuous and monotonic, such that $$f(x)+f(y)=f(z)$$ iif $$f(\lambda x)+f(\lambda y)=f(\lambda z)$$ for all $\lambda>0 $, then $f(x)=ax^b$.
Does anyone know another functional equation that gives a power function as a solution?
On
$$f(x+y)=f(x)+f(y)$$ gives a linear function and $$f(x+y)+f(0)=f(x)+f(y)$$ an affine one.
Then using logarithmic transformations,
$$f(xy)f(1)=f(x)f(y)$$ is your answer.
By the same reasoning, the exponential $ab^x$ is the solution of
$$f(x+y)f(0)=f(x)f(y),$$ and the logarithm $a+\log_bx$ that of
$$f(xy)+f(0)=f(x)+f(y).$$
For all $\ x\ $ and $\ y\ne 0,\ $ the only continuous solutions of the equation $\ f(x)^2 = f(xy)f(x/y)\ $ is $\ f(x) = ax^b.$
An alternative equation similar to Cauchy's functional equation is $\ f(1)f(x\ y) = f(x)f(y)\ $ for all $x,y$.