First off, what do you need to 'prove' time away from home for? The IRS doesn't care if you spent ten dollars on lunch as a business expense 500 miles from home or across the street at the local diner; either way's the same thing; a business expense.
Second, a log book is NOT a record of expenses, receipts are. Keeping receipts for any and every expense in a business (everything from the cost of your health insurance to an evening out with a client or another expediter) is a good idea, but when was the last time your log reflected how much you spent where? Even though you show fueling on a log, you don't show how much fuel you took on and at what price, do you?
Thirdly, I want you to take your log books to the IRS and say "See, I really was in Omaha on the 17th of September. That proves I spent twenty dollars at the Red Lobster." See what they say.
As a sidebar: Do you write off a trip where you met another expediter and talked shop as a business trip? It doesn't matter if you hauled something to get there, just talking to someone can be an 'educational travel' write-off. I have gone on vacation and because I visited a greenhouse and talked to the owner about various things for a couple of hours, (I own a couple of greenhouses and a nursery, but if you just have a garden you could do the same thing) the entire week in OBX (or Hilton Head or wherever) is a write off as a 'business trip'. DO NOT feel guilty in some way or like "I'm not paying my fair share." for doing this; our Congresscritters wrote laws into the most convoluted tax code in all of recorded history to benefit themselves and their donors/friends. They DIDN'T do it for you, but that doesn't mean you shouldn't take advantage of it. In fact, as an American, the concept of an elite ruling class is anathema to the very idea behind this country, and by NOT taking advantage of every tax break you can you contribute to the notion that there are those who are our 'betters' and should be accorded privileges we shouldn't be. Screw 'em and don't worry about keeping a damned log book for seven years; they're not proof of expense in and of themselves, anyway.
PS
I guess you've never done the "One for Monday, two for Tuesday." deal, huh? Remember to use blue ink in one and black in the other so's you don't get 'em confused when the DOT comes to the door.