I want to prove that there are only finitely many rational solutions to $$\left|\frac{\log(5)}{\log(7)}-\frac{a}{b}\right|\le \frac{1}{7^b}$$
And once I have done this, I would like to put a bound on how many solutions there are. Any thoughts?
Edit:
Since for any $x$ and large $b$, $7^b>b^x$, if my equation has infinitely many solutions, then $$\frac{\log(5)}{\log(7)}$$ Must have infinite irrationality measure. Can we prove this isn't true?
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Assuming $a,b$ are integers, it can be verified by brute-force search that if there are any solutions, then $a\ge4000$ or $b\ge2000$.
Checked with Mathematica with following code:
Monitor[Do[If[Abs[Log[5]/Log[7] - a/b] <= 1/7^b, Print[{a, b}]], {b, 1, 1999}, {a, 1, 3999}], {b, a}]
Using Will Jagy's advice, checking continued fractions up to 19 terms no solutions found (more than 19 terms Mathematica overflows).
Do[t = FromContinuedFraction@ContinuedFraction[Log@5/Log@7, a]; If[Abs[Log@5/Log@7 - t] <= 1/7^Denominator@t, Print@a], {a, 1, 19}]
I believe Bakers theorem can put a bound on the number of solutions to this equation.