Exponential/Logarithmic equation system

The system can be written as $$\begin{cases} \log_{10}(2^x)=\log_{10}(11-3^y)\\ \log_{10}(35-(2^{x})^2)=\log_{10}((3^{y})^2) \end{cases}$$ Now if $u:=2^x<\sqrt{35}$ and $v:=3^y<11$, we can throw away the logarithm and solve with respect to $u$ and $v$: $$\begin{cases} u+v=11\\ u^2+v^2=35 \end{cases}$$ Can you take it from here?

P.S. Yes, as you already noted there are no real solutions otherwise $$70=2(u^2+v^2)=(u+v)^2+(u-v)^2\geq(u+v)^2=121.$$

Hint: It is $$2^x=11-3^y$$ and $$35-2^{2x}=3^{2y}$$ Can you proceed?