Evaluate each logarithm by using properties of logarithms and the following facts.
loga(x) = 5.4
  
loga(y) = 1.6
  
loga(z) = 2.4
loga(

y
)


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.

Evaluate each logarithm by using properties of logarithms and the following facts.
loga(x) = 5.4
loga(y) = 1.6
loga(z) = 2.4

Learn how to evaluate logarithmic expressions using properties like the product, quotient, and power rules. This problem utilizes given values for loga(x), loga(y), and loga(z) to demonstrate step-by-step calculations. Suitable for high school students in Grades 9-12.

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