1. We are asked to solve the equation $3^x = 27^x$ using logarithms. 2. First, recognize that $27$ can be written as a power of $3$: $27 = 3^3$. 3. Substitute $27$ with $3^3$ in the equation: $$3^x = (3^3)^x$$ 4. Apply the power of a power rule: $(a^m)^n = a^{mn}$: $$3^x = 3^{3x}$$ 5. Since the bases are the same and the expressions are equal, their exponents must be equal: $$x = 3x$$ 6. Solve for $x$: $$x - 3x = 0$$ $$\cancel{1}x - \cancel{3}x = 0$$ $$-2x = 0$$ 7. Divide both sides by $-2$: $$x = \frac{0}{-2} = 0$$ 8. The solution to the equation is $x = 0$. Final answer: $x = 0$