1. **State the problem:** Solve the equation $3 \cdot 5^{0.2w} = 720$ for $w$. 2. **Isolate the exponential term:** Divide both sides by 3 to get $$5^{0.2w} = \frac{720}{3} = 240.$$ 3. **Use logarithms to solve for $w$:** Taking the natural logarithm (ln) of both sides, $$\ln\left(5^{0.2w}\right) = \ln(240).$$ 4. **Apply the logarithm power rule:** $$0.2w \cdot \ln(5) = \ln(240).$$ 5. **Solve for $w$:** $$w = \frac{\ln(240)}{0.2 \cdot \ln(5)}.$$ 6. **Calculate the numerical value:** Using approximate values $\ln(240) \approx 5.4806$ and $\ln(5) \approx 1.6094$, $$w \approx \frac{5.4806}{0.2 \times 1.6094} = \frac{5.4806}{0.3219} \approx 17.03.$$ **Final answer:** $$w \approx 17.03.$$