CloudWatch has metrics for CloudWatch Logs:
Metrics -> Logs -> Account Metrics
and then you can see the amount of logs coming in by Bytes or Log Events.
If you want to be alerted for the free tier, then you can setup a CloudWatch Alarm and/or an AWS Budget (using the usage not the cost data).
You mentioned that the docs about pricing. Do you mean the CW pricing page? Or the documentation for CloudWatch?
Have you seen this docs page? https://docs.aws.amazon.com/AmazonCloudWatch/latest/monitoring/cloudwatch_billing.html
It's the most comprehensive guide around monitoring, visualizing and optimizing CloudWatch cost and usage, so I suggest to take a look.
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