Example 1 Solve the equation $9 x^{2}-5 x+2=81$.

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Example 1 Solve the equation $9 x^{2}-5 x+2=81$.
Solution 1: Let $t=x^{2}-5 x+2$, then the original equation can be transformed into: $9^{t}=81$, so $t=\log _{9} 81=2$, then $x^{2}-5 x+2=2$, solving yields $x_{1}=0, x_{2}=5$.
Solution 2: Convert the equation to the same base form: $9^{x^{2}-5 x+2}=9^{2}$.

By the monotonicity of the exponential function, we get $x^{2}-5 x+2=2$, solving yields $x_{1}=0, x_{2}=5$.