You've installed Logstash on a RHEL or CentOS box, I presume. Which version?
The bottom line is that for RHEL/CentOS 6.x, Logstash is installed as an upstart service, which you can start with initctl start logstash (or is it initctl logstash start? I never can remember the order).
If you installed on a RHEL/CentOS 7.x system, Logstash will be installed as a systemd service, which you can start with systemctl start logstash.
service start logstash does not work with either, though you can force install a SysV init script by running: /usr/share/logstash/bin/system-install /etc/logstash/startup.options sysv as root (or with sudo). You may still need to go through some hoops to make your system recognize it as a "service" after this. I recommend sticking with upstart/systemd.