I upgraded from Win7 to Win10. Win7 doesn't apply strict permissions to Program Files, but Win10 does. When you try to launch the client, it can't launch because it can't gain write permissions to the log file. You can of course launch the client in admin mode but that shouldn't really be required.
Firstly, I'd argue that the log file is frankly an optional extra that the client should be able to function without.
But secondly, the client should be able to stick it in a temp directory somewhere that is not admin-only, so you can start FAF without administrator permissions.
I would post the log file like the kitty asks but.. well.. given the nature of the issue, I highly doubt that the log file on my disk contains anything relevant.