Cron typically has its own logfile (my distro uses /var/log/cron) and it also may email logs to the cron job owner.
The most important thing to realize when setting up for cron operation is that the cron job itself will not be executing in the warm, comfortable, friendly environment that you'd have when executing the same function as a normal user. So it's best to be careful about what environmental settings you assume are in effect, don't assume you know what your current directory is until (/if) you set it, and access everything via absolute paths. Especially scripts and programs.
In your specific case, you might also be running afoul of the normal mechanisms that keep the /tmp directory clean. I recommend that any log you want to be sure stays around is written to a less volatile location. For system-level cron tasks, the recommended place to put logs is under /var/log. If your service produces more than 1 logfile, make a subdirectory under /var/log to put them all in.
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