Is the any math function that change the power of denominator of an input fraction?

Question

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 $f(r) = r^2$.

Answer 1

Many functions (taking $a$ and $b$ as inputs) can be manufactured. How about $$f(a,b) = log(\frac{a}{b})+ \sum_{n=1}^{k-1}log(\frac{1}{b})$$

Answer 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.

Answer 3

If the input is not $a$ and $b$ but a fraction $r$, you can use continued fractions to find the corresponding $a$ and $b$ such that $r = a/b$, where $a$ and $b$ are relatively prime, and then apply your formula. But as stated by @irchans the obtained function is not continuous