The Maltego Desktop Client's log file can be found in the following directories depending on which operating system you are using:
Windows
C:\Users\<username>\AppData\Roaming\maltego\v<maltego_version>\var\log\
Replace <username> with your username and v<maltego_version> with your current Maltego version, e.g. v4.2.18.
OS X
/Users/<username>/Library/Application Support/maltego/v<maltego_version>/var/log
Replace <username> with your username and v<maltego_version> with your current Maltego version, e.g. v4.2.18.
Linux
~/.maltego/v<maltego_version>/var/log/
Within the Maltego Desktop Client
Alternatively, you can go to Application Menu > Tools > Open Logs Folder.