I have $k^t=8$ and $t^k=9$. The solutions are $2^3=8$ and $3^2=9$ but I Can't think of a way to solve it without knowing the answers. I've tried $t=\frac{ln(8)}{ln(k)}$ and then $(\frac{ln(8)}{ln(k)})^k = 9$ so I can have only one variable. But it didn't work.
I want a method that doesn't lean on "guessing", so I can also use it for large numbers such as $k^t=729$ and $t^k=19683$ (k=9, t=3).
Note that $\ln 8 = 3 \ln 2$, so you can write $ \frac{3\ln 2}{ \ln k} = \sqrt[k] 9$, then $k=2$ is an easy guess. Moreover you can see that the graphs of $y=\frac{3\ln 2}{ \ln k}$ and $y = \sqrt[k] 9$ only have one intersection by monotony.
Probably the problem can not be explicitly solved for random numbers, so you must be happy for the lucky shot in guessing the easy k.
EDIT: I underestimated the verification of monotony, maybe it's not that simple but probably should work something like that.