\(2^x+2^{x+1}+...+2^{x+2021}=2^{2026}-16\)
\(\Rightarrow2^x+2^x.2+...+2^x.2^{2021}=2^{2026}-2^4\)
\(\Rightarrow2^x\left(1+2+...+2^{2021}\right)=2^4\left(2^{2022}-1\right)\)
\(\Rightarrow2^x\left(2^{2022}-1\right)=2^4\left(2^{2022}-1\right)\)
\(\Rightarrow2^x=2^4\)
\(\Rightarrow x=4\)