If you didn't want a LogPlot then this
would seem to do everything you want.
But, since you want a LogPlot, I do not see a way to use that functionality to get the plot you want and things seem to get interesting and more challenging than I would expect the average user to be able to figure out. What follows is my assembling pieces from a variety of different places.
I hope this plots the Planck function for T=3000, 4000 and 5000 Kelvin for wavelengths from .2 to 3 microns, all on top of each other.
In:= h=PlanckConstant/(Joule Second);
planckFunction[t_,x_]:=((2 Pi h c^2)/x^5)/(Exp[h c/k/(x*t)]-1);
Please check this carefully to see if this is correct. It is certainly possible that I have made some mistake in trying to reverse engineer these bits and pieces. Perhaps you can check the results against the non-LogPlot results from BlackBodyProfile and see if you can expose any mistakes when using the temperatures and wavelengths appropriate for your problem.