1. **State the problem:** We want to find the number of days $t$ it takes for the video views $y$ to reach 2500, given the exponential growth function $$y = 80e^{0.2t}.$$ 2. **Write the formula and explain:** The formula for exponential growth is $$y = y_0 e^{kt},$$ where $y_0$ is the initial amount, $k$ is the growth rate, and $t$ is time. Here, $y_0 = 80$ and $k = 0.2$. We want to solve for $t$ when $y = 2500$. 3. **Set up the equation:** $$2500 = 80e^{0.2t}.$$ 4. **Isolate the exponential term:** Divide both sides by 80: $$\frac{2500}{80} = e^{0.2t}$$ $$31.25 = e^{0.2t}.$$ 5. **Take the natural logarithm of both sides:** $$\ln(31.25) = \ln\left(e^{0.2t}\right) = 0.2t.$$ 6. **Solve for $t$:** $$t = \frac{\ln(31.25)}{0.2}.$$ Calculate the value: $$\ln(31.25) \approx 3.442.$$ So, $$t \approx \frac{3.442}{0.2} = 17.21.$$ 7. **Interpretation:** It takes approximately 17.21 days for the video to reach 2500 views. **Final answer:** $$\boxed{17.21 \text{ days}}.$$