1. **State the problem:** We need to order the given mathematical expressions from least to greatest. 2. **Evaluate each expression:** - $\int_1^5 x \, dx = \left[ \frac{x^2}{2} \right]_1^5 = \frac{5^2}{2} - \frac{1^2}{2} = \frac{25}{2} - \frac{1}{2} = \frac{24}{2} = 12$ - $\sum_{i=1}^6 i = 1 + 2 + 3 + 4 + 5 + 6 = 21$ - $\sqrt{2} \approx 1.414$ - $7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040$ - $\frac{6\pi}{3} = 2\pi \approx 6.283$ - $\infty$ is infinite, so it is the greatest. - $e^5 \approx 148.413$ - $\frac{5}{10} = 0.5$ - $\log_4(19)$: Since $4^2=16$ and $4^3=64$, $\log_4(19)$ is between 2 and 3. More precisely, $\log_4(19) = \frac{\ln 19}{\ln 4} \approx \frac{2.944}{1.386} \approx 2.125$ 3. **Order from least to greatest:** $$0.5 < 1.414 < 2.125 < 6.283 < 12 < 21 < 148.413 < 5040 < \infty$$ 4. **Match to original expressions:** $$\frac{5}{10} < \sqrt{2} < \log_4(19) < \frac{6\pi}{3} < \int_1^5 x \, dx < \sum_{i=1}^6 i < e^5 < 7! < \infty$$