I too had some difficulty to do this elgantly. I changed the name of your data file, and commented out the material that didn't work.
idx(1) = find(roundn(cc,-4)==0.1259);
P = polyfit(log10(cc(idx)), log10(H(idx)),1);
cf = polyval(P,log10(cc));
Now try on a linear plot. (Just for testing)
logcc = log10(cc); logH = log10(H);
idx = find(cc<1e-1);
xp = [logcc(idx), fliplr(logcc(idx))];
yp = [cf(idx), fliplr(logH(idx))]
plot(logcc, logH, ...
logcc, cf, '--')
patch(xp, yp, 'y')
Now try, as requested on a log plot. It's certainly not elegant, but does work.
h=patch(10.^xp, 10.^yp, 'y', 'faceAlpha', 0.5)
This gives the following: