Rewrite as a sum or difference of multiples of logarithms.
log Subscript d Baseline left parenthesis x Superscript 7 Baseline y squared z right parenthesis

We are asked to rewrite \( \log_d \left( x^7 y^2 z \right) \) as a sum or difference of multiples of logarithms. 

We will use the following logarithmic properties:

1. **Product Rule**: \( \log_b(AB) = \log_b(A) + \log_b(B) \)
2. **Power Rule**: \( \log_b(A^n) = n \log_b(A) \)

### Step-by-step solution:

Given: 
\[
\log_d \left( x^7 y^2 z \right)
\]

#### Step 1: Apply the product rule
\[
\log_d \left( x^7 y^2 z \right) = \log_d (x^7) + \log_d (y^2) + \log_d (z)
\]

#### Step 2: Apply the power rule
\[
\log_d (x^7) = 7 \log_d (x)
\]
\[
\log_d (y^2) = 2 \log_d (y)
\]

So, the expression becomes:
\[
7 \log_d (x) + 2 \log_d (y) + \log_d (z)
\]

### Final Answer:
\[
\log_d \left( x^7 y^2 z \right) = 7 \log_d (x) + 2 \log_d (y) + \log_d (z)
\]

Would you like further details on logarithmic properties or have any questions?

Here are 5 related questions:
1. How would you rewrite \( \log_d \left( x^5 y^3 z^4 \right) \)?
2. What is the product rule of logarithms, and when is it applied?
3. How can we simplify \( \log_d \left( \frac{x^6}{y^3 z} \right) \)?
4. Can logarithmic expressions be combined into a single logarithm?
5. What are the common logarithmic properties used for solving logarithmic equations?

**Tip**: Always apply the power rule after expanding the product of logarithms to simplify expressions involving exponents.

Rewrite as a sum or difference of multiples of logarithms. log_d(x^7 y^2 z)

Explore how to rewrite the logarithmic expression log_d(x^7 y^2 z) as a sum of multiples of logarithms using the product and power rules. This problem covers key logarithmic properties and is suitable for students in Grades 9-12.

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