Here are the solutions to the problems. 1. Express as a single logarithm: $\log 3 + \log 16 - \log 4 - \log 6$ Step 1: Use the logarithm property $\log a + \log b = \log (a \cdot b)$ to combine the positive terms and the negative terms separately. $$ (\log 3 + \log 16) - (\log 4 + \log 6) $$ $$ \log (3 \times 16) - \log (4 \times 6) $$ $$ \log 48 - \log 24 $$ Step 2: Use the logarithm property $\log a - \log b = \log \left(\frac{a}{b}\right)$ to combine the remaining terms. $$ \log \left(\frac{48}{24}\right) $$ Step 3: Simplify the fraction. $$ \log 2 $$ The expression as a single logarithm is $\boxed{\log 2}$. 2. Simplify: i. $\log_3 63$ Step 1: Find the prime factorization of 63. $$ 63 = 9 \times 7 = 3^2 \times 7 $$ Step 2: Apply the logarithm properties $\log_b (xy) = \log_b x + \log_b y$ and $\log_b x^n = n \log_b x$. $$ \log_3 63 = \log_3 (3^2 \times 7) $$ $$ \log_3 (3^2) + \log_3 7 $$ $$ 2 \log_3 3 + \log_3 7 $$ Step 3: Since $\log_3 3 = 1$, substitute this value. $$ 2(1) + \log_3 7 $$ $$ 2 + \log_3 7 $$ The simplified expression is $\boxed{2 + \log_3 7}$. ii. $\log_3 150$ Step 1: Find the prime factorization of 150. $$ 150 = 15 \times 10 = (3 \times 5) \times (2 \times 5) = 2 \times 3 \times 5^2 $$ Step 2: Apply the logarithm properties $\log_b (xyz) = \log_b x + \log_b y + \log_b z$ and $\log_b x^n = n \log_b x$. $$ \log_3 150 = \log_3 (2 \times 3 \times 5^2) $$ $$ \log_3 2 + \log_3 3 + \log_3 (5^2) $$ $$ \log_3 2 + \log_3 3 + 2 \log_3 5 $$ Step 3: Since $\log_3 3 = 1$, substitute this value. $$ \log_3 2 + 1 + 2 \log_3 5 $$ The simplified expression is $\boxed{1 + \log_3 2 + 2 \log_3 5}$.