You’ve got the expression:

$4\log_4 y - \frac{1}{2}\log_4 z + 7\log_4 x$

We can simplify this by applying the properties of logarithms:

Properties of logarithms we can use:

1. Power Rule: $\log_b (a^n) = n \log_b a$
2. Logarithmic Addition/Subtraction: $a \log_b c = \log_b(c^a)$

Simplification:

1. For the term $4\log_4 y$, we apply the power rule: $4\log_4 y = \log_4 y^4$

2. For the term $\frac{1}{2}\log_4 z$, we again apply the power rule: $\frac{1}{2}\log_4 z = \log_4 z^{\frac{1}{2}} = \log_4 \sqrt{z}$

3. For the term $7\log_4 x$, apply the power rule: $7\log_4 x = \log_4 x^7$

Putting it all together:

$\log_4 y^4 - \log_4 \sqrt{z} + \log_4 x^7$

Now, use the property $\log_b a - \log_b c = \log_b \frac{a}{c}$ and $\log_b a + \log_b c = \log_b (a \cdot c)$ to combine the terms:

$= \log_4 \left( \frac{y^4 \cdot x^7}{\sqrt{z}} \right)$

Final simplified form:

$\log_4 \left( \frac{y^4 x^7}{\sqrt{z}} \right)$

Let me know if you'd like further clarification!

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Here are some related questions to explore:

1. How do you solve logarithmic equations that involve both addition and subtraction?
2. Can the expression be further simplified if we change the base of the logarithms?
3. What are the steps for solving exponential equations using logarithms?
4. How do the properties of logarithms change when the base is different from 4?
5. How can you apply logarithms to solve real-world problems, like population growth or compound interest?

Tip: Always remember the properties of logarithms (power rule, product rule, and quotient rule) to simplify complex logarithmic expressions.