\(5^{x+1}-5^x=500\)
\(\Rightarrow5^x\cdot5^1-5^x=500\)
\(\Rightarrow5^x\cdot5-5^x=500\)
\(\Rightarrow5^x\cdot\left(5-1\right)=500\)
\(\Rightarrow5^x\cdot4=500\)
\(\Rightarrow5^x=500:4\)
\(\Rightarrow5^x=125\)
\(\Rightarrow5^x=5^3\)
\(\Rightarrow x=3\)