1. **State the problem:** Solve the equation $4^{3x} = 8x - 3$ for $x$. 2. **Analyze the equation:** The left side is an exponential function $4^{3x}$ and the right side is a linear function $8x - 3$. 3. **Rewrite the exponential base:** Note that $4 = 2^2$, so $$4^{3x} = (2^2)^{3x} = 2^{6x}.$$ 4. **Rewrite the equation:** $$2^{6x} = 8x - 3.$$ 5. **Consider the domain:** The right side $8x - 3$ must be positive because the left side $2^{6x}$ is always positive. So, $$8x - 3 > 0 \implies x > \frac{3}{8} = 0.375.$$ 6. **Check for possible solutions:** This equation is transcendental (exponential equals linear), so exact algebraic solutions are difficult. We can check for solutions by testing values or using numerical methods. 7. **Test $x=1$:** $$2^{6(1)} = 2^6 = 64,$$ $$8(1) - 3 = 5,$$ Not equal. 8. **Test $x=2$:** $$2^{12} = 4096,$$ $$8(2) - 3 = 13,$$ Not equal. 9. **Test $x=0.5$:** $$2^{3} = 8,$$ $$8(0.5) - 3 = 1,$$ Not equal. 10. **Test $x=0.4$:** $$2^{2.4} \approx 5.278,$$ $$8(0.4) - 3 = 0.2,$$ Not equal. 11. **Test $x=0.6$:** $$2^{3.6} \approx 12.125,$$ $$8(0.6) - 3 = 1.8,$$ Not equal. 12. **Conclusion:** The exponential grows much faster than the linear function, and the two sides do not appear to intersect for $x > 0.375$. For $x \leq 0.375$, the right side is negative or zero, but the left side is positive, so no solution there. 13. **Check $x=0$:** $$2^0 = 1,$$ $$8(0) - 3 = -3,$$ No equality. 14. **Check $x$ near 0.375:** At $x=0.375$, right side is zero, left side is $2^{2.25} \approx 4.76$, no equality. 15. **Graphical or numerical methods are needed for exact solution.** 16. **Summary:** No real solution satisfies $4^{3x} = 8x - 3$ because the exponential is always positive and grows faster than the linear, and the linear is negative or zero for $x \leq 0.375$. **Final answer:** No real solution exists for the equation $4^{3x} = 8x - 3$.