1. Problem: Solve the exponential equation $10^x = 100^{2x - 3}$ by finding a common base. 2. Recall that $100 = 10^2$, so rewrite the right side: $$10^x = (10^2)^{2x - 3}$$ 3. Use the power of a power rule: $(a^m)^n = a^{mn}$: $$10^x = 10^{2(2x - 3)} = 10^{4x - 6}$$ 4. Since the bases are the same and nonzero, set the exponents equal: $$x = 4x - 6$$ 5. Solve for $x$: $$x - 4x = -6$$ $$\cancel{1}x - \cancel{4}x = -6$$ $$-3x = -6$$ 6. Divide both sides by $-3$: $$x = \frac{-6}{-3} = 2$$ 7. Final answer: $x = 2$