Why my fit for a logarithm function looks so wrong

I'm plotting this dataset and making a logarithmic fit, but, for some reason, the fit seems to be strongly wrong, at some point I got a good enough fit, but then I re ploted and there were that bad fit. At the very beginning there were a 0.0 0.0076 but I changed that to 0.001 0.0076 to avoid the asymptote.

I'm using (not exactly this one for the image above but now I'm testing with this one and there is that bad fit as well) this for the fit

f(x) = a*log(k*x + b)
fit = fit f(x) 'R_B/R_B.txt' via a, k, b

And the output is this

Also, sometimes it says 7 iterations were as is the case shown in the screenshot above, others only 1, and when it did the "correct" fit, it did like 35 iterations or something and got a = 32 if I remember correctly

Edit: here is again the good one, the plot I got is this one. And again, I re ploted and get that weird fit. It's curious that if there is the 0.0 0.0076 when the good fit it's about to be shown, gnuplot says "Undefined value during function evaluation", but that message is not shown when I'm getting the bad one.

Do you know why do I keep getting this inconsistence? Thanks for your help

As I already mentioned in comments the method of fitting antiderivatives is much better than fitting derivatives because the numerical calculus of derivatives is strongly scattered when the data is slightly scatered.

The principle of the method of fitting an integral equation (obtained from the original equation to be fitted) is explained in https://fr.scribd.com/doc/14674814/Regressions-et-equations-integrales . The application to the case of y=a.ln(c.x+b) is shown below.

Numerical calculus :

In order to get even better result (according to some specified criteria of fitting) one can use the above values of the parameters as initial values for iterarive method of nonlinear regression implemented in some convenient software.

NOTE : The integral equation used in the present case is :

NOTE : On the above figure one can compare the result with the method of fitting an integral equation to the result with the method of fitting with derivatives.

Acknowledgements : Alex Sveshnikov did a very good work in applying the method of regression with derivatives. This allows an interesting and enlightening comparison. If the goal is only to compute approximative values of parameters to be used in nonlinear regression software both methods are quite equivalent. Nevertheless the method with integral equation appears preferable in case of scattered data.

UPDATE (After Alex Sveshnikov updated his answer)

The figure below was drawn in using the Alex Sveshnikov's result with further iterative method of fitting.

The two curves are almost indistinguishable. This shows that (in the present case) the method of fitting the integral equation is almost sufficient without further treatment.

Of course this not always so satisfying. This is due to the low scatter of the data.

In ADDITION , answer to a question raised in comments by CosmeticMichu :