1. **State the problem:** Find the values of the logarithms: a) $\log_2 1024$ b) $\log_2 \frac{1}{4}$ c) $\log_4 8$ 2. **Recall the logarithm definition and properties:** - $\log_b a = c$ means $b^c = a$. - Important rules: - $\log_b (xy) = \log_b x + \log_b y$ - $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$ - Change of base formula: $\log_b a = \frac{\log_k a}{\log_k b}$ for any base $k$. 3. **Calculate each part:** a) $\log_2 1024$ - Since $1024 = 2^{10}$, - $\log_2 1024 = \log_2 2^{10} = 10$. b) $\log_2 \frac{1}{4}$ - Note $\frac{1}{4} = 4^{-1} = (2^2)^{-1} = 2^{-2}$, - So $\log_2 \frac{1}{4} = \log_2 2^{-2} = -2$. c) $\log_4 8$ - Express 8 and 4 as powers of 2: $8 = 2^3$, $4 = 2^2$, - Using change of base: $\log_4 8 = \frac{\log_2 8}{\log_2 4} = \frac{3}{2} = 1.5$. **Final answers:** a) 10 b) -2 c) 1.5 These results align with the properties of logarithms and the given triangles where the hypotenuse represents sums or differences of logs.