Not necessarily . Some carriers have info printed on their company logs that they require and would not be on a standard triplicate log . Some carriers have a recap and DVIR attached to the logs that must be filled out . I drive for a tank carrier and dates of several required tank inspections must be shown on each daily log .
Usually different companies, or at least the companies worth their salt, have their own logs already printed up rather than those general use logs bought at the truck stop. In that instance when working for 2 carriers all that needs to be done is provide a copy of each companies logs to the other company. In this case it may be very advantageous to have a double-copy log (3-page log). But the problem with working for 2 carriers is you must figure all the time from both carriers into the 11, 14, and 60/70 requirements e.g. if you drive for carrier "a" for 10 hours and had 2-hours on line 4, then all you can work for carrier "b" is drive for 1-hour B4 the 11 and 14-hour rules becomes effective. You can't drive 11-hours for both carriers. Same thing for the 60/70.
I had a personal log book to use before i was hired on,,,, this was to log bobtailing to wash the truck, repairs, getting it home from the dealer etc. I think my ? is answered and ill just use the co log book ,,, which is what i did any way,, It just happened to be the right way.