If you as an ordinary user decide to run a program, the natural place for its logs are in your home directory. Your home directory is meant for you to store all your files, whether they are logs of a program you run or anything else.
If your program is meant to be run by ordinary users and you want to set a default location, that default location should be in a “dot directory”, i.e. a directory whose name begins with a . in the user's home directory. An application called myapp typically places its state files under .myapp. (Certain types of files can go in different directories, for example .config/myapp for configuration files (that only change from user intervention) or .cache/myapp (for files that can be removed without hurting anything except performance), but there's no such directory for logs.).
The XDG specification suggests ${XDG_STATE_HOME}/myapp as the directory for log files for an application called myapp run by a human user on their own account. ${XDG_STATE_HOME} defaults to ~/.local/state.
If the program is executed as part of the system, running as a typically dedicated system user, then the natural place for its logs is in /var/log. Create a subdirectory /var/log/myapp and give it appropriate permissions so that your application can write there.
If relevant and your operating system allows it, mark the log file as append-only. Only root can do this. This has the advantage that if your application is compromised, it won't be able to erase past logs, which can be very useful for forensic analysis of the compromise. You will need root's intervention to rotate the log: chown so that the log file is can no longer be opened by the application, rename the log file, create a new append-only file with appropriate ownership, then notify the application to open the new empty file.
You can make any application log to the system logs by calling logger(1) or syslog(3).