I don't know how to determine the cause of the error.

First, read down the error message.
First, the error 5xx series is issuing the error status code on the server side.
If you didn't issue it yourself, you'd most likely see a script stop error.

Next is

2016-07-2023:46:07 Error: MissingControllerException Controller class supportedBrowsersController could not be found.

Regarding this content,
It may occur if you connect to a path that does not exist, such as reading a file.
The log file seems to have been deleted, so
I tried to write some kind of log, but I couldn't because it was deleted.
Or I tried to load it, but the file didn't exist.
It's just a guess, but it's possible.

Also, the line that suggests the stop point described below

0/var/www/html/site name/webroot/index.php(110)—Dispatcher ->dispatch (Object(CakeRequest), Object(CakeResponse))

Above,
in line 110 of index.php Dispatcher->dispatch(Object(CakeRequest),Object(CakeResponse))
indicates that the has failed and stopped.

Are you trying to log a connected request or something from index.php?
Either way, I don't have the source code, so I can't say exactly what caused it.
index.php is the only way to follow the code?

If the IO of the file fails, it might work if you reinstall the empty log file.