📝 Add async logger lifecycle API docs

docs/api/async-logger-last-error.md

@@ -0,0 +1,76 @@
+---
+name: async-logger-last-error
+group: api
+category: async
+update-time: 20260512
+description: Read the last error recorded by the async logger worker during runtime failure handling.
+key-word:
+    - async
+    - logger
+    - failure
+    - public
+---
+
+## Async-logger-last-error
+
+Read the last error string recorded by the async logger worker. This helper gives the textual detail behind `has_failed()`.
+
+### Interface
+
+```moonbit
+pub fn[S] AsyncLogger::last_error(self : AsyncLogger[S]) -> String {}
+```
+
+#### input
+
+- `self : AsyncLogger[S]` - Async logger whose last worker error should be inspected.
+
+#### output
+
+- `String` - Last recorded worker error message, or an empty string if no error was recorded.
+
+### Explanation
+
+Detailed rules explaining key parameters and behaviors
+
+- `run()` resets the stored error string when the worker starts.
+- If the worker loop fails, the error text is captured from the raised exception.
+- An empty string normally means no failure has been recorded.
+- This helper reports worker execution errors, not ordinary overflow or backpressure conditions.
+
+### How to Use
+
+Here are some specific examples provided.
+
+#### When Need Failure Detail In Diagnostics
+
+When a compact error message should be surfaced to operators:
+```moonbit
+if logger.has_failed() {
+  println(logger.last_error())
+}
+```
+
+In this example, the error string is only read when failure is present.
+
+#### When Export Full Async State
+
+When custom diagnostics want to include the last error field directly:
+```moonbit
+let err = logger.last_error()
+```
+
+In this example, the helper provides the textual failure detail without building a full snapshot.
+
+### Error Case
+
+e.g.:
+- If no runtime failure has occurred, the method returns an empty string.
+
+- If callers need broader context than just the error text, they should use `state()`.
+
+### Notes
+
+1. Read this helper together with `has_failed()` when interpreting worker health.
+
+2. The stored value is a diagnostic string, not a typed error object.