Does gps make any log, which is store in there positioning system ?
Even if gps is off in our smart phone and we don't have any sim card in it and our internet is off too still is their any chance to get our location?
No, there isn't any log on the satellite - it's just broadcasting a time signal and an identifier. Anyone can pick it up - think of it as being like a radio broadcast.
Phone location, as in how police track a specific phone, doesn't work using GPS directly. A limited amount of location data can be obtained by looking at which cell tower the phone is connected to. The range that this provides varies in relation to the cell size - in dense areas, like cities, it gives a more precise location than in sparse coverage areas, like lightly populated rural regions. If the phone has an internet connection, you can also plot location with some accuracy from the WiFi access points which can be seen (especially in countries with regular Google maps updates). If the internet connection is off, that probably won't work, but it might be possible to identify locations which have been visited at a later point, when internet connectivity is restored. It would depend on the specific applications on the phone.
For question 1, Matthew answer is perfect, with nothing to add.
For question 2, the answer is no, but only if the phone is in the airline more (not just SIM card is out). Please note that the phone can still connect to cell towers even without a SIM card, because the Emergency number could be dialed without a valid SIM card. Of course GPS and Wi-Fi should be off too.