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The given problem asks to expand and simplify the logarithmic expression \( \log_5 \left( \frac{z}{25} \right)^3 \).

Let's break it down step-by-step.

### Step 1: Use the Power Rule of Logarithms
The power rule states that \( \log_b (M^n) = n \log_b (M) \). So, we can move the exponent 3 in front of the logarithm:
\[
\log_5 \left( \frac{z}{25} \right)^3 = 3 \log_5 \left( \frac{z}{25} \right)
\]

### Step 2: Use the Quotient Rule of Logarithms
The quotient rule states that \( \log_b \left( \frac{M}{N} \right) = \log_b (M) - \log_b (N) \). Now we apply this to the expression inside the logarithm:
\[
\log_5 \left( \frac{z}{25} \right) = \log_5 (z) - \log_5 (25)
\]

### Step 3: Simplify the Logarithms
We know that \( \log_5 (25) = \log_5 (5^2) \). Using the power rule again, we get:
\[
\log_5 (25) = 2 \log_5 (5) = 2
\]
because \( \log_5 (5) = 1 \).

### Step 4: Combine the Results
Substitute the simplified value of \( \log_5 (25) \) back into the expression:
\[
3 \left( \log_5 (z) - 2 \right)
\]

### Step 5: Distribute the 3
Now distribute the factor of 3 across the terms:
\[
3 \log_5 (z) - 6
\]

### Final Answer:
The expanded and simplified expression is:
\[
3 \log_5 (z) - 6
\]

Would you like further explanation or more details? Here are some related questions you might find helpful:

1. What are the other logarithmic properties, and how are they applied?
2. How can we simplify logarithmic expressions with variables?
3. How do we handle logarithms with different bases?
4. What are the real-world applications of logarithmic functions?
5. Can we solve logarithmic equations using these properties?

**Tip**: When working with logarithms, always remember the three key rules: product, quotient, and power rules. They simplify most expressions easily!

Expand and simplify the logarithmic expression log_5((z/25)^3).

This problem explores how to expand and simplify the logarithmic expression log_5((z/25)^3) using the power and quotient rules of logarithms. Suitable for students in Grades 9-12, the solution involves applying key logarithmic properties step-by-step.

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