If $\log\left(\left(\frac{b}{a}\right)^{\frac{b}{3}}\right) + \log\left(\left(\sqrt[3]{\frac{a}{b}}\right)^{9a}\right)=1$, then what is $a^2+b^2$?

Question

Let $a$ and $b$ be two positive integers where $b$ is a multiple of $a$. If $\log\left(\left(\frac{b}{a}\right)^{\frac{b}{3}}\right) + \log\left(\left(\sqrt[3]{\frac{a}{b}}\right)^{9a}\right)=1$then what is $a^2+b^2$?

Answer 1

$b = na$

$b/3 (\log b - \log a) + 3a (\log a - \log b) = 1$

$(na/3 - 3a)(\log n + \log a - \log a) = 1$

$a (n/3 - 3) \log n = 1$

$n = 10, a = 3, b = 30$ is a solution

you can see that you won't find more solution in integers (e.g. n = 10^k)