ApplicationDeployment.CurrentDeployment.ActivationUri might work
"A zero-length string if the TrustUrlParameters property in the deployment manifest is false, or if the user has supplied a UNC to open the deployment or has opened it locally. Otherwise, the return value is the full URL used to launch the application, including any parameters."
BUT what I think you really want is ApplicationDeployment.CurrentDeployment.DataDirectory which gives you a folder you can write data to. When you update the application anyways you'll lose what was in the original .exe folder, but you can migrate the data directory over to a new version of the app. Your app can write to this folder with whatever log files it has - and I'm pretty sure its guaranteed to be writable.