My KDE is broken, the desktop Widgets cannot be displayed. And many KDE programs look like this:
All the software written in GTK is all right.
Can any one give me any suggestion about what's happening and which part is broken?
Where can I find error log?
You don't say what disto you're using (please add it to the tags in your question), nor how you got in this state - new installation? dist-upgrade? hard crash? That would be useful information.
But in general, here's what I would try:
Log out and go to a bare terminal with CTRL-ALT-F1, then login and rename your .kde / .kde4 directories: mv ~/.kde ~/.kde.old and reboot or otherwise restart your GUI system.
That will at least tell you whether the problem is messed up personal settings or messed-up system files.
If that doesn't fix the issue (that directory will be recreated when you start up KDE again; all your settings will be lost, but you can recover them - carefully, one by one - from the backup you just made), then I would first try sudo apt-get check (assuming you're on a Debian-based distro).
If that doesn't report any problems, then I would update my system - possibly even do a dist-upgrade without changes any sources.
If this issue still wasn't fixed, I would run sudo dpkg -l > ~/Desktop/dpkg_out.txt to get a list of installed or uninstalled packages and their state in a file, and then look through the file for problems as explained here.
Finally, if all of that failed, I would take a good hard look at my video drivers.
Good luck!
