Which directory is used to put rendered PNG file?

The directories used are controlled on the filter settings page of the filter with DruTeX enabled.

I checked the filtering setting pages and have enabled 0777 mode to that directory. On the filtering page, I can see "The temporary directory is writable. The image directory is writable." at the Debugging mode section.

But I still got the error saying "The selected file /forrest/www/drupal/files/tmp/file5oZnfb could not be copied." I have enabled the debug mode. Can you tell me where is the log file?

I also got these errors:
Executing "latex" failed. Either LaTeX isn't installed or the "latex" executable is not in one of the search paths.
You cannot render high-quality images without LaTeX.
Mimetex maybe an alternative for you. Consult the documentation on how to use Mimetex.
Executing "dvipng" failed. Either dvipng isn't installed or the "dvipng" executable is not in one of the search paths.
You cannot use dvipng for rendering images (check the conversion method above).
Executing "convert" (ImageMagick) failed. Either ImageMagick isn't installed or the "convert" executable is not in one of the search paths.
You cannot use ImageMagick for rendering images (check the conversion method above).

I have tried to just type them in the shell terminal, and they can be found. I don't know why DruTex said that it can't find them.

I have enabled the debug mode. Can you tell me where is the log file?

Its goes into the standard watchdog log.

I have tried to just type them in the shell terminal, and they can be found. I don't know why DruTex said that it can't find them.

The php exec command doesn't have all the environment variables that your shell has. Try setting the render method to custom, and using full paths to the executables.