/tmp/pupdial.log is updated after every line of output and can be viewed in wvdial connection log.
When I try to achieve a similar thing for the volume monitor, the output is full-buffered i.e. the output is not flushed after every line. Data gets written only when it has written about 1kb. None of the following work like I want.
pup-volume-monitor &> "$LOG_FILE" #forewards stderr only
pup-volume-monitor 2>&1 > "$LOG_FILE" #forewards stderr only