1. **State the problem:** Solve the equation $$6^x = \frac{12^x}{8}$$ for $x$. 2. **Rewrite the equation:** Multiply both sides by 8 to eliminate the denominator: $$8 \cdot 6^x = 12^x$$ 3. **Express bases as powers of 2 and 3:** $$6 = 2 \times 3$$ $$12 = 2^2 \times 3$$ So, $$8 \cdot (2 \times 3)^x = (2^2 \times 3)^x$$ 4. **Rewrite the equation using exponents:** $$8 \cdot 2^x \cdot 3^x = 2^{2x} \cdot 3^x$$ 5. **Express 8 as a power of 2:** $$8 = 2^3$$ So, $$2^3 \cdot 2^x \cdot 3^x = 2^{2x} \cdot 3^x$$ 6. **Combine powers of 2 on the left:** $$2^{3+x} \cdot 3^x = 2^{2x} \cdot 3^x$$ 7. **Divide both sides by $3^x$ to cancel:** $$\frac{2^{3+x} \cdot \cancel{3^x}}{\cancel{3^x}} = \frac{2^{2x} \cdot \cancel{3^x}}{\cancel{3^x}}$$ Which simplifies to: $$2^{3+x} = 2^{2x}$$ 8. **Since the bases are equal, set exponents equal:** $$3 + x = 2x$$ 9. **Solve for $x$:** $$3 = 2x - x$$ $$3 = x$$ **Final answer:** $$x = 3$$