Example 2 Solve the equation $5^{2 x}-6 \cdot 5^{x}+5=0$.

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Example 2 Solve the equation $5^{2 x}-6 \cdot 5^{x}+5=0$.
Let $5^{x}=y$, then the original equation can be transformed into: $y^{2}-6 y+5=0$, solving this yields $y_{1}=5, y_{2}=1$, so:
When $y_{1}=5$, $5^{x}=5$, which means $x=1$;
When $y_{2}=1$, $5^{x}=1$, which means $x=0$.
Therefore, the solutions to the original equation are $x_{1}=1, x_{2}=0$.