Suggesting method for removing noise and image restoration?
The question is: "Suggest the method for removing noise with transforming $g$ and filter this image in the transformed domain." I believe it is more about "finding a method" than "doing the job practically".

Your first step is quite sound: remove the power complexity. But you are left with a product of a noise and a ($\log$) image. Most well known tools operate in a linear way. So use your hypotheses a little further: since $f&gt;1$, its logarithm is positive. hence $v\log f$ is positive too. So you can turn the noise/signal product into a sum, by applying a logarithm again:

$$ \log \log g = \log v + \log \log f\,.$$

Now  $ \log v$ can be considered as an additive noise to $\log \log f$. With this linearized model, you can try get insights on the nature of the noise, looking at flat regions of the image, checking histograms, etc. Once done, 
you can  apply your best filter (adapted to the statistics of $ \log v$, yet unknown), and $\exp \cdot \exp$ the result. The function $v\to \exp  \exp v$ grows very fast (from rechneronline.de):



So it relatively flattens low values with respect to higher values, with can result in  contrast unbalance. This effect could be counterbalanced by intensity bounding, nonlinear mappings, etc., but more importantly  by proper image restoration.

For your information, using linear filters in a $\log$ domain is an instance  of homomorphic filtering, which your solution may belong too:


  Homomorphic filtering is a generalized technique for signal and image
  processing, involving a nonlinear mapping to a different domain in
  which linear filter techniques are applied, followed by mapping back
  to the original domain