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Umm...doesn't the formula imply that the dog age starts at 31?

That makes no sense either. My 1 year old dog is definitely not a 31 year old adult as far as behavior goes (incidentally, I am myself ~31,).

I'd put 1 year approximately equivalent to a teenager, depending on the size of the breed (12 year old for larger breeds, late teens for small breeds)

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Not quite. Ages are usually counted from 0. ln(0) is undefined, but after that from the first moment after birth until some time in its 52nd day, a dog's age is negative in dog-time (that's probably why it's considered important for the puppy to remain with its parents during this period; a negative-aged dog on its own is like a naked singularity). Then from day 52 to the end of its first year, the dog will age from 0 to 31.

The other end of the scale is surprising too: 200 human years is less than 116 dog years.

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