I don't see anywhere in your code where you're saving the backend NotesDocument object after appending text to the rich text field.
After calling rti.Update save the Doc object. You will need to close the uidocument and then re-open the backend document:
Dim wksp As New NotesUIWorkspace
Dim uidoc As NotesUIDocument
Dim Db As NotesDatabase
Dim doc As NotesDocument
Dim rti As NotesRichTextItem
Dim NoteID As String
Set uidoc = wksp.CurrentDocument
Set doc = uidoc.Document
Set Db = doc.ParentDatabase
Set rti = doc.GetFirstItem("theLog")
rti.Appendtext "Something to be added to the log"
rti.update
Call doc.Save(True, True)
Delete rti
NoteID = doc.NoteID
Call uidoc.Close(True)
If Not uidoc Is Nothing Then
Delete uidoc
End If
If Not doc Is Nothing Then
Delete doc
End If
Set doc = Db.GetDocumentById(NoteID)
If Not doc Is Nothing Then
Call wksp.EditDocument(True, doc, False)
End If