6) \( \log _{2}\left(8 \cdot 3^{6}\right)^{3} \)

To solve \( \log _{2}\left(8 \cdot 3^{6}\right)^{3} \), we start by simplifying the expression inside the logarithm. First, \( \left(8 \cdot 3^{6}\right)^{3} \) can be expanded as: \[ (8^3) \cdot (3^6)^3 \] Calculating \(8^3\): \[ 8 = 2^3 \implies 8^3 = (2^3)^3 = 2^{9} \] Calculating \((3^6)^3\): \[ (3^6)^3 = 3^{18} \] Now, we can combine these results: \[ (8 \cdot 3^{6})^{3} = 2^{9} \cdot 3^{18} \] Now we have: \[ \log _{2}\left(2^{9} \cdot 3^{18}\right) \] Using the logarithm property \(\log_b(x \cdot y) = \log_b(x) + \log_b(y)\), we can separate the terms: \[ \log _{2}(2^{9}) + \log _{2}(3^{18}) \] Calculating the first term: \[ \log _{2}(2^{9}) = 9 \] For the second term, we use the property \(\log_b(x^n) = n \cdot \log_b(x)\): \[ \log _{2}(3^{18}) = 18 \cdot \log _{2}(3) \] Putting it all together, we have: \[ 9 + 18 \cdot \log _{2}(3) \] Therefore, the final answer is: \[ \log _{2}\left(8 \cdot 3^{6}\right)^{3} = 9 + 18 \cdot \log _{2}(3) \]