First of all are you sure you want to log data to text file? May be it will be better to store log into separate table ?
If you want to work with text file:
The Windows process spawned by xp_cmdshell has the same security rights as the SQL Server service account.
Check the security rights for this account.
xp_cmdshell can be enabled and disabled by using the Policy-Based Management or by executing sp_configure.
Check you have it enabled.
EXEC sp_configure 'show advanced options', 1
GO
RECONFIGURE
GO
EXEC sp_configure 'xp_cmdshell', 1
GO
RECONFIGURE
GO
When it is called by a user that is not a member of the sysadmin fixed server role, xp_cmdshell connects to Windows by using the account name and password stored in the credential named ##xp_cmdshell_proxy_account##. If this proxy credential does not exist, xp_cmdshell will fail.
You need to create proxy account.
EXEC sp_xp_cmdshell_proxy_account [MyDomain\SQLServerProxy], 'usdcu&34&23'
Add permissions to use this SP:
USE master;
GRANT EXECUTE on xp_cmdshell to Current_user
Here is a more detailed information.