Getting the logo on the screen is actually a positive sign.
It proves that the screen is working, and also that the power is probably OK as well. It also proves that the firmware is partially working.
If possible try re-installing the firmware into the TV and see if this solves the problem. It may not work as sometimes you cannot install the same version only an updated version.
Here is a link to the support page for your TV. Scroll down to Software and Firmware to find the two files necessary. The guide will tell you how to install the firmware and the firmware itself. Note that it is a .zip file and will have to be uncompressed but most PCs do that automatically these days. Also even though it might say so, make sure that you load the firmware into the root directory of the USB drive that you plug into the TV. Do not put it into a folder as it won't be found.
Follow the directions and be patient as it might take a little while to complete. I cannot say how long as it varies between brands, but generally it is not long.
Hopefully this is of some help.