Step-by-step Solution:

We will solve each part step by step. Part (a): Evaluating \( \log_4 (\log_3 81) \) First, simplify \( \log_3 81 \): \[ \log_3 81 = \log_3 (3^4) = 4 \] Now, evaluate \( \log_4 4 \): \[ \log_4 4 = 1 \] Thus, \( A = III \). \[ \boxed{A - III} \] Part (b): Solving for \( k \) in \( 3^{4\log_9 7} = 7^k \) Rewriting the logarithm: \[ \log_9 7 = \frac{\log_3 7}{\log_3 9} = \frac{\log_3 7}{2} \] Thus, \[ 3^{4 \cdot \frac{\log_3 7}{2}} = 7^k \] \[ 3^{2\log_3 7} = 7^k \] \[ (3^{\log_3 7})^2 = 7^k \] \[ 7^2 = 7^k \] \[ k = 2 \] Thus, \( B = IV \). \[ \boxed{B - IV} \] Part (c): Evaluating \( 2^{\log_3 5} - 5^{\log_3 2} \) Using the property: \[ a^{\log_b c} = c^{\log_b a} \] We get: \[ 2^{\log_3 5} = 5^{\log_3 2} \] So, \[ 2^{\log_3 5} - 5^{\log_3 2} = 0 \] Thus, \( C = I \). \[ \boxed{C - I} \] Part (d): Evaluating \( \log_2 [\log_2 (256)] \) First, simplify \( \log_2 256 \): \[ \log_2 256 = \log_2 (2^8) = 8 \] Now, evaluate \( \log_2 8 \): \[ \log_2 8 = \log_2 (2^3) = 3 \] Thus, \( D = II \). \[ \boxed{D - II} \] Final Matching: \[ A - III, \quad B - IV, \quad C - I, \quad D - II \] This corresponds to option D. \[ \boxed{\text{Option D}} \]