a. \( { }^{2} \log 6+{ }^{2} \log 4-{ }^{2} \log 3 \) b. \( { }^{3} \log 3 \sqrt{3}-{ }^{3} \log 2 \sqrt{3}+{ }^{3} \log 54 \)

To simplify the logarithmic expressions, we will use the properties of logarithms, including the change of base formula and the properties of logarithms such as \( \log a + \log b = \log(ab) \) and \( \log a - \log b = \log\left(\frac{a}{b}\right) \). ### Part a: \( { }^{2} \log 6 + { }^{2} \log 4 - { }^{2} \log 3 \) Using the property of logarithms: \[ { }^{2} \log 6 + { }^{2} \log 4 = { }^{2} \log (6 \cdot 4) = { }^{2} \log 24 \] Now, we can combine this with the subtraction: \[ { }^{2} \log 24 - { }^{2} \log 3 = { }^{2} \log \left(\frac{24}{3}\right) = { }^{2} \log 8 \] Since \( 8 = 2^3 \), we can express this as: \[ { }^{2} \log 8 = { }^{2} \log (2^3) = 3 \cdot { }^{2} \log 2 \] Thus, the simplified expression for part a is: \[ 3 \cdot { }^{2} \log 2 \] ### Part b: \( { }^{3} \log 3 \sqrt{3} - { }^{3} \log 2 \sqrt{3} + { }^{3} \log 54 \) First, we simplify \( { }^{3} \log 3 \sqrt{3} \) and \( { }^{3} \log 2 \sqrt{3} \): \[ { }^{3} \log 3 \sqrt{3} = { }^{3} \log (3^{1} \cdot 3^{1/2}) = { }^{3} \log (3^{3/2}) = \frac{3}{2} \cdot { }^{3} \log 3 \] \[ { }^{3} \log 2 \sqrt{3} = { }^{3} \log (2 \cdot 3^{1/2}) = { }^{3} \log 2 + { }^{3} \log 3^{1/2} = { }^{3} \log 2 + \frac{1}{2} { }^{3} \log 3 \] Now substituting these into the original expression: \[ \frac{3}{2} { }^{3} \log 3 - \left( { }^{3} \log 2 + \frac{1}{2} { }^{3} \log 3 \right) + { }^{3} \log 54 \] Combining the terms: \[ \left( \frac{3}{2} { }^{3} \log 3 - \frac{1}{2} { }^{3} \log 3 \right) - { }^{3} \log 2 + { }^{3} \log 54 \] This simplifies to: \[ { }^{3} \log 3 - { }^{3} \log 2 + { }^{3} \log 54 \] Now, we can combine the logarithms: \[ { }^{3} \log \left( \frac{3 \cdot 54}{2} \right) \] Calculating \( \frac{3 \cdot 54}{2} \): \[ 3 \cdot 54 = 162 \quad \text{and} \quad \frac{162}{2} = 81 \] Thus, we have: \[ { }^{3} \log 81 \] Since \( 81 = 3^4 \), we can express this as: \[ { }^{3} \log (3^4) = 4 \cdot { }^{3} \log 3 \] So the simplified expression for part b is: \[ 4 \cdot { }^{3} \log 3 \] ### Final Answers: a. \( 3 \cdot { }^{2} \log 2 \) b. \( 4 \cdot { }^{3} \log 3 \)