📝 Add async logger composition API docs

docs/api/async-logger-child.md

@@ -0,0 +1,78 @@
+---
+name: async-logger-child
+group: api
+category: async
+update-time: 20260512
+description: Derive a child async logger by composing the current target with a child target segment.
+key-word:
+    - async
+    - logger
+    - target
+    - public
+---
+
+## Async-logger-child
+
+Create a child async logger by composing the current target with another target segment. This is the standard API for hierarchical async logger naming such as `app.worker` or `service.http.client`.
+
+### Interface
+
+```moonbit
+pub fn[S] AsyncLogger::child(self : AsyncLogger[S], target : String) -> AsyncLogger[S] {}
+```
+
+#### input
+
+- `self : AsyncLogger[S]` - Parent async logger whose target should be extended.
+- `target : String` - Child target segment or suffix.
+
+#### output
+
+- `AsyncLogger[S]` - A new async logger whose default target is the composed child path.
+
+### Explanation
+
+Detailed rules explaining key parameters and behaviors
+
+- If the parent target is empty, the child target becomes the full target.
+- If the child target is empty, the parent target is preserved.
+- If both are non-empty, they are joined with `.`.
+- Queue settings, sink wiring, and runtime behavior are preserved in the returned logger.
+
+### How to Use
+
+Here are some specific examples provided.
+
+#### When Need Hierarchical Async Targets
+
+When subsystem logs should stay grouped under one async namespace:
+```moonbit
+let logger = async_logger(console_sink(), target="service")
+let worker = logger.child("worker")
+```
+
+In this example, `worker` emits under `service.worker` while keeping the same async queue behavior.
+
+#### When Build Deep Async Scopes Step By Step
+
+When deeper target composition should remain readable:
+```moonbit
+let http = async_logger(console_sink(), target="app")
+  .child("http")
+  .child("client")
+```
+
+In this example, the final logger target becomes `app.http.client`.
+
+### Error Case
+
+e.g.:
+- If `target` is empty, the returned logger keeps the original parent target.
+
+- If callers need complete replacement rather than composition, `with_target(...)` should be used instead.
+
+### Notes
+
+1. This is the preferred API for hierarchical async logger naming.
+
+2. Composition changes the target only and does not rebuild the queue or sink.