Logging script

Copy to the ThisWorkbook code module.

Private Sub Workbook_Open()
    Dim lRow As Long
    Dim c As Range
    
    With Worksheets("Log")
        Set c = .Cells.Find(Application.UserName)
        If Not c Is Nothing Then
            lRow = c.Row
        Else
            lRow = .Cells(.Rows.Count, 1).End(xlUp).Row + 1
        End If
        
        .Cells(lRow, 1) = "Opened"
        .Cells(lRow, 2) = Application.UserName
        .Cells(lRow, 3) = Format(Now, "dd-mm-yyyy hh:mm:ss")
    End With
End Sub

Private Sub Workbook_BeforeClose(Cancel As Boolean)
    Dim lRow As Long
    Dim c As Range
    
    With Worksheets("Log")
        Set c = .Cells.Find(Application.UserName)
        If Not c Is Nothing Then
            lRow = c.Row
        Else
            lRow = .Cells(.Rows.Count, 1).End(xlUp).Row + 1
        End If
        
        .Cells(lRow, 1) = "Closed"
        .Cells(lRow, 2) = Application.UserName
        .Cells(lRow, 3) = Format(Now, "dd-mm-yyyy hh:mm:ss")
    End With
    ThisWorkbook.Save
End Sub