One Way Function - How to Prove? Function $f$ is a length shortening function. It reduces to log of size of input, i.e $|f(x)|/\log(|x|) \leq$ $a\ positive\ constant$. Is this a one way function?

Edit: $f$ is any function. Unknown definition, only constraint on the ratio being constant for all $x$
F(x) = 1 matches your requirement and obviously isn't one way. It is trivial to find a pre image for this function as any value is a pre image of 1.
One way function doesn't mean it's hard to find the original value. It means it's hard to find any pre image which leads to the requested output value.
Finding the original value of a compression function is obviously impossible to do with high confidence if the original data is chosen uniformly. This for information reasons regardless of computing power.