Livy exposes REST API in 2 ways: session and batch. In your case, since we assume you are not using session, you are submitting using batches. You can post your batch using the curl command:
curl http://livy-server-IP:8998/batches
Once you have submitted the job, you would get the batch ID in return. Then you can curl using the command:
curl http://livy-server-IP:8998/batches/{batchId}/log
You can find the documentation at:
https://livy.incubator.apache.org/docs/latest/rest-api.html
If you want to avoid the above steps, you can use a ready-made AMI (namely, LightningFLow) from AWS Marketplace which provides Airflow with a custom Livy operator. Livy operator submits and tracks the status of the job every 30 sec (configurable), and it also provides spark logs at the end of the spark job in Airflow UI logs.
Note: LightningFlow comes pre-integrated with all required libraries, Livy, custom operators, and local Spark cluster.
Link for AWS Marketplace:
https://aws.amazon.com/marketplace/pp/Lightning-Analytics-Inc-LightningFlow-Integrated-o/B084BSD66V
This will enable you to view consolidated logs at one place, instead of shuffling between Airflow and EMR/Spark logs (Ambari/Resource Manager).