see this, Dunham, William - Euler - The Master of Us All - (1999), page 17. It may help

7 is prime not a factor of 2, 3 and 10.

That is why you can not use the properties of LOGARITHM to obtain a pattern for base b.

But it is still solvable, Euler's Formula, log (base b) of 2 = (ln 2)/(ln b)