You have a PHP error in your child theme which is causing that. If you don’t see an error in the apache log, it might be that you PHP errors are stored at some other location, or just not being reported at all. Check your PHP error settings in your php.ini file.
Anyway, most likely your problem is caused by the code in your child theme’s functions.php file. If you just copied it from the parent theme you have a problem with redeclaration of function. Try to rename it and see if you don’t get the 500 error anymore.
If you did copy the functions.php file then you are the doing it wrong as the parent theme’s functions.php file is being loaded as well,m therefor the functions in it are going to be executed and available to be called from your code