After fixing the many typos you have (what is log? Mathematica uses Log), and adding a constraint on k
else complex solution will result, here it is
data = {{0.617, 0.8}, {0.605, 0.6}, {0.5997, 0.4}, {0.5972, 0.2}, {0.5985, 0.1}};
soln = NonlinearModelFit[data, {y - 2478.82*w Log[k c + 1], k > 0},
{y, w, k}, c, MaxIterations -> 1000]

There is still some warning about converge and the tolerance. You can play with options to try to eliminate these.
reference: problem-with-nonlinearmodelfit-the-function-value-is-not-a-list-of-real-num
y
,w
,k
, and the error variance) with just 5 data points. You've got to be kidding! You need more data or a much, much simpler model or tell whoever has asked you to do this that it shouldn't be done. $\endgroup$ – JimB Dec 12 '16 at 3:56