1. **State the problem:** We want to understand how to go from the equation $$(\ln 3)(\ln 3 + \ln x) = (\ln 4)(\ln 4 + \ln x)$$ to the next step $$(\ln 3 - \ln 4) \ln x = (\ln 4)^2 - (\ln 3)^2.$$ 2. **Recall the distributive property:** When you have expressions like $a(b+c)$, you multiply $a$ by each term inside the parentheses: $$a(b+c) = ab + ac.$$ 3. **Apply the distributive property to both sides:** $$ (\ln 3)(\ln 3) + (\ln 3)(\ln x) = (\ln 4)(\ln 4) + (\ln 4)(\ln x).$$ This simplifies to: $$ (\ln 3)^2 + (\ln 3)(\ln x) = (\ln 4)^2 + (\ln 4)(\ln x).$$ 4. **Group like terms:** Move all terms involving $\ln x$ to one side and constants to the other: $$ (\ln 3)(\ln x) - (\ln 4)(\ln x) = (\ln 4)^2 - (\ln 3)^2.$$ 5. **Factor out $\ln x$ on the left side:** $$ (\ln 3 - \ln 4) \ln x = (\ln 4)^2 - (\ln 3)^2.$$ This is exactly the step from the 3rd to the 4th line. **Summary:** We used the distributive property to expand both sides, then rearranged terms and factored $\ln x$ to isolate it. Final answer: $$ (\ln 3 - \ln 4) \ln x = (\ln 4)^2 - (\ln 3)^2.$$