[Task] 회원-time log

### ISSUE
- Group: `client`, `server`, `sr`
- Type: `bug`, `feature`, `delete`
- Detail: 첫 로그인 이후 홈에서 수업으로 이동한 유저의 time log API

### TODO
1. [ ] time log API

### Estimated time
> Pick one
### `1h`

### Labels
- Estimated time: `E: 1h`
- Group : `client`
- Sprint: `Sprint__NUMBER__`
- Urgency: `Low`

No files, tests, or entry points are named. Start by tracing the client flow from the first login through the home screen to class navigation, then inspect existing API request patterns. Done means the time log API is implemented for that transition and the request is made at the intended point.

Written by the indexing model from the issue text.