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.