This should get you started.
On opening your file and not yet present a sheet will be created called Logfile as last sheet in your workbook.
Then UserName and Date/Time will be logged, the sheet protected and the workbook saved.
Just before closing your file Date/Time will be logged and the workbook saved again.
Code:
Dim lrow As Long

Private Sub Workbook_Open()
If Worksheets(Worksheets.Count).Name <> "Logfile" Then
    Worksheets.Add after:=Worksheets(Worksheets.Count)
    Worksheets(Worksheets.Count).Name = "Logfile"
    With Sheets("Logfile")
        .Range("A1").Resize(, 3) = Split("UserName|Last opened|Last closed", "|")
        .Columns("A").ColumnWidth = 20
        .Columns("B:C").ColumnWidth = 18
        .Range("B:C").NumberFormat = "mm-dd-yyyy h:mm:ss"
    End With
End If
With Sheets("Logfile")
    .Protect Password:="Rajesh", UserInterfaceOnly:=True
    lrow = .Range("A" & Rows.Count).End(xlUp).Row + 1
    .Range("A" & lrow).Value = Environ("username")
    .Range("B" & lrow).Value = Now()
End With
ActiveWorkbook.Save
End Sub

Private Sub Workbook_BeforeClose(Cancel As Boolean)
If Worksheets(Worksheets.Count).Name = "Logfile" Then
    Worksheets("Logfile").Range("C" & lrow).Value = Now()
End If
ActiveWorkbook.Save
End Sub