if y = 2^x then taking logs of both x and y we have
log y = 2 ^ log x, still an exponential relation.
so it depends on whether "log enough scale" applies to both axes...
Then there's functions like n -> 2^2^...^2 with an exponential tower of height n. That's doesn't become linear with any finite number of logs...
If y = 2^x, then log y != 2^log x because of Jensen's inequality. You can take log of both sides to keep the equality, but applying the transformation inside of 2^() results in the equality turning into a strict inequality.
I'd say if the graph isn't linear, one of the axes aren't log enough...