📝 Add async logger lifecycle API docs

docs/api/async-logger-dropped-count.md

@@ -0,0 +1,77 @@
+---
+name: async-logger-dropped-count
+group: api
+category: async
+update-time: 20260512
+description: Read the number of records dropped by the async logger because of overflow or queue clearing.
+key-word:
+    - async
+    - logger
+    - queue
+    - public
+---
+
+## Async-logger-dropped-count
+
+Read the cumulative number of records dropped by an async logger. This metric is useful when overflow policy or forced shutdown behavior may discard queued data.
+
+### Interface
+
+```moonbit
+pub fn[S] AsyncLogger::dropped_count(self : AsyncLogger[S]) -> Int {}
+```
+
+#### input
+
+- `self : AsyncLogger[S]` - Async logger whose dropped-record counter should be inspected.
+
+#### output
+
+- `Int` - Number of records dropped so far.
+
+### Explanation
+
+Detailed rules explaining key parameters and behaviors
+
+- The counter increases when overflow policy discards records.
+- The counter can also increase when `close(clear=true)` or `shutdown(clear=true)` abandons queued records.
+- This is a cumulative counter for the lifetime of the logger value.
+- Use this helper when you need a focused loss metric rather than a full `state()` snapshot.
+
+### How to Use
+
+Here are some specific examples provided.
+
+#### When Need To Detect Loss Under Pressure
+
+When queue overflow may discard records:
+```moonbit
+if logger.dropped_count() > 0 {
+  println("async log loss detected")
+}
+```
+
+In this example, the code checks whether the async logger has already discarded records.
+
+#### When Validate Shutdown Behavior In Tests
+
+When a test intentionally clears the queue:
+```moonbit
+logger.close(clear=true)
+ignore(logger.dropped_count())
+```
+
+In this example, the drop counter helps confirm that abandoned backlog was tracked.
+
+### Error Case
+
+e.g.:
+- If no records have been dropped, the method simply returns `0`.
+
+- If callers need to know why records were lost, they must interpret this metric together with overflow policy and shutdown behavior.
+
+### Notes
+
+1. This helper reports record loss only; it does not explain the full logger state.
+
+2. Pair it with `pending_count()` or `state()` when investigating backlog pressure.