The fft can be reversed almost exactly (using ifft in Matlab). Any numerical errors will be tiny.
The log-polar transformation cannot usually be reversed exactly, because you would need to massively oversample in the central part of the pattern in order to avoid undersampling in the outer part. This means that the reverse transform generally involves interpolation in the peripheral parts of the image, and it is therefore approximate. For an example, see
http://www.cogs.susx.ac.uk/users/davidy/log_lines/index.html
(If you need functions to carry out the log-polar transform and its approximate inverse, you could download this file exchange submission:
http://www.mathworks.com/matlabcentral/fileexchange/27023
It needs the Image Processing Toolbox.)
