If your log method is an async method like this:
public async Task SendLog(Log mylog)
{
// some async logic.
}
You can simply call it with:
SendLog(log).ConfigureAwait(false);
The send process will continue and will not block your current thread.
Reference: https://stackoverflow.com/a/53184241/8676371
But if you want to handle the exception asynchronously, you could do:
SendLog(log).
ContinueWith(t => Console.WriteLine(t.Exception), TaskContinuationOptions.OnlyOnFaulted);
This will allow you to deal with an exception on a thread other than the "main" thread. This means you don't have to "wait" for the call SendLog() from the original thread; but, still allows you to do something with an exception--but only if an exception occurs.
Reference: https://stackoverflow.com/a/15524271/8676371