app/log: fix race condition making the log test fail

[KaloyanTanev]

Fix flaky test
https://github.com/ObolNetwork/charon/actions/runs/23892947707/job/69670537788?pr=4425

category: test
ticket: none

[sonarqubecloud]

Quality Gate passed

Issues
0 New issues
0 Accepted issues

Measures
0 Security Hotspots
0.0% Coverage on New Code
0.0% Duplication on New Code

See analysis details on SonarQube Cloud

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 56.48%. Comparing base (8a1dce1) to head (5f0211e).
⚠️ Report is 1 commits behind head on main.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #4441      +/-   ##
==========================================
- Coverage   56.50%   56.48%   -0.02%     
==========================================
  Files         245      245              
  Lines       32608    32608              
==========================================
- Hits        18426    18420       -6     
- Misses      11837    11838       +1     
- Partials     2345     2350       +5     

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