The given expression is:

$\log_b(2x) - 3\log_b(4x) - 5\log_b(x)$

We need to combine this expression into a single logarithm using logarithmic properties.

Step-by-step approach:

1. Apply the power rule: The logarithmic power rule states that $a\log_b(x) = \log_b(x^a)$. Let's apply this to the terms with coefficients:

   • $\log_b(2x)$ remains as it is.
   • $3\log_b(4x) = \log_b((4x)^3) = \log_b(64x^3)$
   • $5\log_b(x) = \log_b(x^5)$

   Now the expression becomes: $\log_b(2x) - \log_b(64x^3) - \log_b(x^5)$

2. Apply the subtraction rule: The subtraction rule of logarithms states that $\log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right)$. We can now combine the terms step by step.

   First, combine the first two terms: $\log_b\left(\frac{2x}{64x^3}\right) = \log_b\left(\frac{2}{64} \cdot \frac{x}{x^3}\right) = \log_b\left(\frac{1}{32} \cdot \frac{1}{x^2}\right) = \log_b\left(\frac{1}{32x^2}\right)$

   Now, include the third term: $\log_b\left(\frac{1}{32x^2}\right) - \log_b(x^5) = \log_b\left(\frac{\frac{1}{32x^2}}{x^5}\right) = \log_b\left(\frac{1}{32x^7}\right)$

Thus, the final expression as a single logarithm is:

$\log_b\left(\frac{1}{32x^7}\right)$

Final Answer:

$\log_b\left(\frac{1}{32x^7}\right)$

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

Here are 5 related questions to explore:

1. What are the properties of logarithms?
2. How does the power rule for logarithms work in other contexts?
3. Can logarithms be applied to non-algebraic problems, such as real-world scenarios?
4. How can we simplify logarithmic expressions involving division?
5. What is the significance of the base in logarithms?

Tip: When simplifying logarithms, always look for opportunities to apply the power and quotient rules to reduce the number of terms.