You can refer to the logic app pricing document for more details. You will only be charged for the resources that it creates/use.
As @Ken Tucker mentioned you can utilize the pricing page or calculator to calculate the estimated cost of your resource.
You can review the Data retention pricing. All the inputs and outputs that are stored in your logic app's run history get billed based on a logic app's run retention period. Logic apps that run in an ISE don't incur data retention costs.
If you are using the Azure REST API then you should not be charged for calling the management API's.