1. **State the problem:** Solve for $x$ in the equation $$\log_5(x + 22) - \log_5(x - 2) = 2.$$\n\n2. **Recall the logarithm subtraction rule:** \n$$\log_b A - \log_b B = \log_b \left(\frac{A}{B}\right)$$\nThis means we can rewrite the left side as a single logarithm:\n$$\log_5 \left(\frac{x + 22}{x - 2}\right) = 2.$$\n\n3. **Rewrite the logarithmic equation in exponential form:** \nSince $\log_5 y = 2$ means $y = 5^2$, we have:\n$$\frac{x + 22}{x - 2} = 5^2 = 25.$$\n\n4. **Solve the resulting equation:** \nMultiply both sides by $x - 2$ to clear the denominator:\n$$x + 22 = 25(x - 2).$$\n\n5. **Expand and simplify:** \n$$x + 22 = 25x - 50.$$\n\n6. **Isolate $x$ terms:** \n$$x + 22 - 25x = -50$$\n$$\cancel{1}x - 25x = -50 - 22$$\n$$-24x = -72.$$\n\n7. **Divide both sides by $-24$:** \n$$x = \frac{-72}{-24} = 3.$$\n\n8. **Check the domain:** \nThe arguments of the logarithms must be positive:\n$$x + 22 > 0 \Rightarrow 3 + 22 = 25 > 0,$$\n$$x - 2 > 0 \Rightarrow 3 - 2 = 1 > 0.$$\nBoth conditions hold, so $x=3$ is valid.\n\n**Final answer:** $$x = 3.$$