Write to custom log file from a Bash script

In Linux, I know how to write a simply message to the /var/log/messages file, in a simple shell script I created:

#!/bin/bash
logger "have fun!"

I want to stop throwing messages into the default /var/log/messages file, and create my own.

I tried this:

#!/bin/bash
logger "have more fun" > /var/log/mycustomlog

It still logs to /var/log/messages. It did create the /var/log/mycustomlog, but it's empty.

Anyone see what I'm missing?