1. **State the problem:** Find the smallest number $n$ such that $160^n > 80$. 2. **Formula and approach:** To solve inequalities involving exponents, we use logarithms. The inequality $160^n > 80$ can be rewritten using logarithms as: $$n > \frac{\log(80)}{\log(160)}$$ 3. **Calculate the logarithms:** - Calculate $\log(80)$ - Calculate $\log(160)$ 4. **Evaluate the right side:** $$n > \frac{\log(80)}{\log(160)}$$ Using common logarithms (base 10): - $\log(80) \approx 1.9031$ - $\log(160) \approx 2.2041$ So, $$n > \frac{1.9031}{2.2041} \approx 0.8633$$ 5. **Interpretation:** Since $n$ must be greater than approximately $0.8633$, the smallest integer $n$ satisfying the inequality is: $$n = 1$$ **Final answer:** The smallest number $n$ such that $160^n > 80$ is $1$.