$ docker run -it -p 1883:1883 -v "$PWD/mosquitto/config:/mosquitto/config" -v "$PWD/mosquitto/data:/mosquitto/data" -v "$PWD/mosquitto/log:/mosquitto/log" eclipse-mosquitto

When running the image, the default configuration values are used. To use a custom configuration file, create your mosquitto.conf in $PWD/mosquitto/config/mosquitto.conf, then mount the config directory to /mosquitto/config.

$ docker run -it -p 1883:1883 -v "$PWD/mosquitto/config:/mosquitto/config" eclipse-mosquitto
Configuration can be changed to:

persist data to /mosquitto/data
log to /mosquitto/log/mosquitto.log
i.e. add the following to mosquitto.conf:

persistence true
persistence_location /mosquitto/data/
log_dest file /mosquitto/log/mosquitto.log