1. The first problem is to evaluate $\log_4(17)$. This means finding the exponent $x$ such that $4^x = 17$. 2. The formula to convert logarithms to a common base is: $$\log_a b = \frac{\log_c b}{\log_c a}$$ where $c$ is any positive number (commonly 10 or $e$). 3. Using base 10 logarithms: $$\log_4(17) = \frac{\log_{10}(17)}{\log_{10}(4)}$$ 4. Calculate the values: $$\log_{10}(17) \approx 1.2304$$ $$\log_{10}(4) \approx 0.6021$$ 5. Divide the two values: $$\log_4(17) = \frac{1.2304}{0.6021} \approx 2.044$$ --- 6. The second problem is the summation $\sum_{i=1}^3 i$ which means adding the integers from 1 to 3. 7. Calculate the sum: $$1 + 2 + 3 = 6$$ --- 8. The third problem is the definite integral $\int_2^7 x \, dx$. 9. The integral of $x$ is: $$\frac{x^2}{2}$$ 10. Evaluate the definite integral: $$\int_2^7 x \, dx = \left[ \frac{x^2}{2} \right]_2^7 = \frac{7^2}{2} - \frac{2^2}{2} = \frac{49}{2} - \frac{4}{2} = \frac{45}{2} = 22.5$$ --- **Final answers:** - $\log_4(17) \approx 2.044$ - $\sum_{i=1}^3 i = 6$ - $\int_2^7 x \, dx = 22.5$