1. The problem is to solve the equation $$(\lg x)^2 - 5 \lg x + 6 = 0.$$\n2. Let us use a substitution to simplify the problem. Set $$y = \lg x.$$ Then the equation becomes $$y^2 - 5y + 6 = 0.$$\n3. This is a quadratic equation in $y$. We can factor it as $$(y-2)(y-3)=0.$$\n4. So, the solutions for $y$ are $$y=2 \quad \text{or} \quad y=3.$$\n5. Recall that $y = \lg x$, so we have $$\lg x = 2 \quad \text{or} \quad \lg x = 3.$$\n6. To solve for $x$, rewrite the logarithm equations in exponential form: $$x = 10^2 = 100,$$ and $$x = 10^3 = 1000.$$\n7. Therefore, the solutions to the original equation are $$x=100 \quad \text{and} \quad x=1000.$$