1. **State the problem:** Solve for $x$ in the equation $$16(3)^x = 81(2)^x.$$\n\n2. **Rewrite constants as powers:** Note that $16 = 2^4$ and $81 = 3^4$. Substitute these into the equation:\n$$2^4 \cdot 3^x = 3^4 \cdot 2^x.$$\n\n3. **Rewrite the equation:**\n$$2^4 \cdot 3^x = 3^4 \cdot 2^x.$$\n\n4. **Divide both sides by $2^x 3^x$ to isolate terms:**\n$$\frac{2^4 \cdot 3^x}{2^x \cdot 3^x} = \frac{3^4 \cdot 2^x}{2^x \cdot 3^x}.$$\n\n5. **Simplify the fractions using cancellation:**\n$$\frac{\cancel{3^x} \cdot 2^4}{2^x \cdot \cancel{3^x}} = \frac{3^4 \cdot \cancel{2^x}}{\cancel{2^x} \cdot 3^x}$$\nwhich simplifies to\n$$\frac{2^4}{2^x} = \frac{3^4}{3^x}.$$\n\n6. **Rewrite the equation as:**\n$$2^{4 - x} = 3^{4 - x}.$$\n\n7. **Since bases 2 and 3 are different and positive, the only way for the equality to hold is if the exponents are equal and the bases are equal, which is impossible. So the only solution is when the exponents make both sides equal. This happens if:**\n$$4 - x = 0,$$\nwhich gives\n$$x = 4.$$\n\n8. **Check the solution:** Substitute $x=4$ back into the original equation:\n$$16 \cdot 3^4 = 81 \cdot 2^4,$$\nCalculate each side:\n$$16 \cdot 81 = 81 \cdot 16,$$\nwhich is true.\n\n**Final answer:**\n$$x = 4.$$