Is the any math function that change the power of denominator of an input fraction? I'm seeking for a function that gets a fraction as input and change its denominator power in the log function like follow:

$f(\frac{a}{b}) = log(\frac{a}{b^k})$

Is it possible to find a function like f?

p.s: The input of function is not a and b, it's $\frac{a}{b}$ and I need the definition of function, like $f(r) = r^2$.
You can define a function as follows.  Let $f(a/b) := \log(a/b^2)$ where $a$ and $b$ are relatively prime positive integers.  The function $f$ "maps" positive rational numbers (i.e. fractions) to real numbers.  Note that $f$ is not smooth or continuous.
Many functions (taking $a$ and $b$ as inputs) can be manufactured. How about $$f(a,b) = log(\frac{a}{b})+ \sum_{n=1}^{k-1}log(\frac{1}{b})$$