What is the default location for MSBuild logs?

Log file from Visual Studio is only supported for C++ projects. You just have to work with the output window for others.

See this similar thread: VS2010: minimal build log in output and detailed log in log file

And in case you happen to do this for a C++ project, the file is at:

... build log in the intermediate files directory ... The path and name of the build log is represented by the MSBuild macro expression, $(IntDir)\$(MSBuildProjectName).log.

Use build output instead of logging to file. Instead of copy/paste, simply click somewhere in the output and press CTRL + S to save. Visual Studio will prompt you for a location (tested with Visual Studio 2017, but I'm assuming this works in earlier versions too).