Let's work on solving each of the parts (a), (b), and (c) of the problem step by step. Here's the problem:

Simplify each of the following expressions, where $x > 0$ and $x \neq 1$.

(a)

$\frac{\log x^5 + 4\log x}{\log x^3}$

Solution:

1. Use the logarithmic power rule: $\log x^n = n\log x$.

   • $\log x^5 = 5\log x$
   • $\log x^3 = 3\log x$

   So, the numerator becomes:
   $5\log x + 4\log x = 9\log x$

   The denominator becomes: $3\log x$

2. Simplify the fraction:
   $\frac{9\log x}{3\log x} = \frac{9}{3} = 3$

Thus, the simplified expression is: $\boxed{3}$

(b)

$\log_2 x - 3\log_8 x$

Solution:

1. Convert $\log_8 x$ into base 2 using the change of base formula:
   $\log_a b = \frac{\log_c b}{\log_c a}$

   Here, $\log_8 x = \frac{\log_2 x}{\log_2 8}$. Since $8 = 2^3$, $\log_2 8 = 3$.
   Therefore: $\log_8 x = \frac{\log_2 x}{3}$

2. Substitute this into the original expression:
   $\log_2 x - 3\log_8 x = \log_2 x - 3\left(\frac{\log_2 x}{3}\right)$

3. Simplify: $\log_2 x - \log_2 x = 0$

Thus, the simplified expression is: $\boxed{0}$

(c)

$\frac{\log_{125} x^3}{\log_{25} \sqrt[3]{x}}$

Solution:

1. Apply the logarithmic power rule:

   • $\log_{125} x^3 = 3\log_{125} x$
   • $\sqrt[3]{x} = x^{1/3}$, so $\log_{25} \sqrt[3]{x} = \frac{1}{3}\log_{25} x$

   The expression becomes: $\frac{3\log_{125} x}{\frac{1}{3}\log_{25} x}$

2. Simplify the fraction: $\frac{3}{\frac{1}{3}} \cdot \frac{\log_{125} x}{\log_{25} x} = 9 \cdot \frac{\log_{125} x}{\log_{25} x}$

3. Use the change of base formula to express $\log_{125} x$ and $\log_{25} x$ in base 5:

   • $125 = 5^3$, so $\log_{125} x = \frac{\log_5 x}{\log_5 125} = \frac{\log_5 x}{3}$
   • $25 = 5^2$, so $\log_{25} x = \frac{\log_5 x}{\log_5 25} = \frac{\log_5 x}{2}$

   The fraction becomes: $\frac{\log_{125} x}{\log_{25} x} = \frac{\frac{\log_5 x}{3}}{\frac{\log_5 x}{2}} = \frac{1}{3} \cdot \frac{2}{1} = \frac{2}{3}$

4. Substitute back into the expression: $9 \cdot \frac{2}{3} = 6$

Thus, the simplified expression is: $\boxed{6}$

Final Answers:

(a) $\boxed{3}$
(b) $\boxed{0}$
(c) $\boxed{6}$

Would you like further details, or explanations on specific steps?

Here are 5 related questions to explore:

1. What is the change of base formula and how is it applied in logarithms?
2. How do you simplify expressions involving logarithmic powers?
3. What are the conditions for using the logarithmic properties, such as the power rule?
4. How can you convert between logarithms of different bases?
5. What are some practical applications of logarithmic simplifications in real-life contexts?

Tip: Always double-check the base of the logarithm when applying simplification rules, as different bases lead to different results!