Q2: Use properties of logarithms to expand the expression \( \log \left(\frac{x^{4} z^{2}}{100 y^{3}}\right) \) as much as possible.

To expand the expression \( \log \left(\frac{x^{4} z^{2}}{100 y^{3}}\right) \) using properties of logarithms, we can apply the following properties: 1. The logarithm of a quotient: \[ \log \left(\frac{a}{b}\right) = \log(a) - \log(b) \] 2. The logarithm of a product: \[ \log(ab) = \log(a) + \log(b) \] 3. The logarithm of a power: \[ \log(a^b) = b \log(a) \] Now, we can apply these properties step by step: 1. Start with the quotient: \[ \log \left(\frac{x^{4} z^{2}}{100 y^{3}}\right) = \log(x^{4} z^{2}) - \log(100 y^{3}) \] 2. Expand \( \log(x^{4} z^{2}) \) using the product property: \[ \log(x^{4} z^{2}) = \log(x^{4}) + \log(z^{2}) \] 3. Now, apply the power property: \[ \log(x^{4}) = 4 \log(x) \quad \text{and} \quad \log(z^{2}) = 2 \log(z) \] So, \[ \log(x^{4} z^{2}) = 4 \log(x) + 2 \log(z) \] 4. Next, expand \( \log(100 y^{3}) \): \[ \log(100 y^{3}) = \log(100) + \log(y^{3}) \] 5. Again, apply the power property: \[ \log(y^{3}) = 3 \log(y) \] And since \( 100 = 10^2 \), we have: \[ \log(100) = \log(10^2) = 2 \log(10) = 2 \] (assuming the logarithm is base 10). 6. Therefore, \[ \log(100 y^{3}) = 2 + 3 \log(y) \] Now, putting it all together: \[ \log \left(\frac{x^{4} z^{2}}{100 y^{3}}\right) = (4 \log(x) + 2 \log(z)) - (2 + 3 \log(y)) \] 7. Finally, simplify: \[ \log \left(\frac{x^{4} z^{2}}{100 y^{3}}\right) = 4 \log(x) + 2 \log(z) - 2 - 3 \log(y) \] Thus, the expanded expression is: \[ \log \left(\frac{x^{4} z^{2}}{100 y^{3}}\right) = 4 \log(x) + 2 \log(z) - 3 \log(y) - 2 \]