SQL Server 2012 Browser Service doesn't appear
The Log you posted seems incomplete, also it is not the log for SQL Server installation but for patch update. You can see in the log 

Requested action:   Patch 


Anyways log is helpful because it shows below

Product features discovered:
  Product              Instance             Instance ID                    Feature                                  Language             Edition              Version         Clustered 
  SQL Server 2012                                                          Management Tools - Basic                 1033                 Express Edition      11.0.2100.60    No        
  SQL Server 2012                                                          LocalDB                                  1033                 Express Edition      11.1.3000.0     No        


Whenever you run patch SQL Server writes in Log what all instances and features it has found on the current system and as per log you only installed


SQL Server management studio
Local DB not Database engine.


You need to download ENU\x64\SQLEXPR_x64_ENU.exe because your system is 64 bit from This Link

I am not sure error is because of missing Browser or missing Express Database Engine service. Browser service is installed with SQL Express database engine so if you are looking to connect to database engine not Local DB you need to to install Express DB engine.