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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?

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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.

Write the expression using a single logarithm: log_b(2x) - 3log_b(4x) - 5log_b(x)

Simplify the expression log_b(2x) - 3log_b(4x) - 5log_b(x) into a single logarithm using logarithmic properties. The solution demonstrates the power rule and quotient rule of logarithms, making it suitable for high school students studying advanced algebra and logarithmic operations.

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