My lecturer gave us a potential bonus to the grade if we can fully solve those equations: $y^x=9 $ $x^y=8$
easy to see that the value of $x$ and $y$ are $2$ & $3$ but what is the correct full way of showing the solution for that?
appreciate your help geniuses <3
$y^x=9 \Rightarrow x=\frac{ln9}{lny}$ , $x^y=8 \Rightarrow y=\frac{ln8}{lnx}$ Substitute to get $x=\frac{ln9}{ln(\frac{ln8}{lnx})}=\frac{ln9}{ln(ln8)-ln(lnx)} \Rightarrow x(ln(ln8)-ln(lnx))=ln9 \Rightarrow xln(ln8)-xlnln(x)=ln9$ And then well, using Wolframe Alpha to solve the above equation, we get $x=2$. This then implies $2^y=8 \Rightarrow y=3$.
Hope that help!