To evaluate expressions involving logarithms, we can use the given facts:

• $\log_a(x) = 5.4$
• $\log_a(y) = 1.6$
• $\log_a(z) = 2.4$

Properties of Logarithms:

1. Product Rule: $\log_a(mn) = \log_a(m) + \log_a(n)$
2. Quotient Rule: $\log_a\left(\frac{m}{n}\right) = \log_a(m) - \log_a(n)$
3. Power Rule: $\log_a(m^n) = n \cdot \log_a(m)$

Now, you want to evaluate the following logarithmic expressions based on these facts:

1. Evaluate $\log_a(xy)$: Using the product rule: $\log_a(xy) = \log_a(x) + \log_a(y) = 5.4 + 1.6 = 7.0$

2. Evaluate $\log_a\left(\frac{x}{z}\right)$: Using the quotient rule: $\log_a\left(\frac{x}{z}\right) = \log_a(x) - \log_a(z) = 5.4 - 2.4 = 3.0$

3. Evaluate $\log_a(x^2)$: Using the power rule: $\log_a(x^2) = 2 \cdot \log_a(x) = 2 \cdot 5.4 = 10.8$

4. Evaluate $\log_a\left(\frac{z}{y^2}\right)$: Using the quotient and power rules: $\log_a\left(\frac{z}{y^2}\right) = \log_a(z) - \log_a(y^2) = \log_a(z) - 2 \cdot \log_a(y)$ Substituting the values: $\log_a(z) - 2 \cdot \log_a(y) = 2.4 - 2 \cdot 1.6 = 2.4 - 3.2 = -0.8$

Do you need any more details or have any questions?

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Here are 5 additional related questions:

1. How would you evaluate $\log_a\left(x^3z\right)$?
2. Can you calculate $\log_a\left(\frac{x^2}{y^3}\right)$?
3. What is the value of $\log_a(y^4)$ using the power rule?
4. How would you approach evaluating $\log_a\left(\frac{xz}{y^2}\right)$?
5. Can you simplify and solve $\log_a\left(x^3y^2\right)$?

Tip: Always remember to apply the power rule first when there are exponents inside the logarithmic function, as it simplifies the expression early.