To express $\log_b\left(\frac{p^2q^6}{m^5b^9}\right)$ in terms of sums and differences of logarithms, use the properties of logarithms. Here's the simplified expression:

$\log_b\left(\frac{p^2q^6}{m^5b^9}\right) = \log_b(p^2) + \log_b(q^6) - \log_b(m^5) - \log_b(b^9)$

Now simplify further using the power rule of logarithms ($\log_b(x^n) = n \cdot \log_b(x)$):

$\log_b(p^2) = 2\log_b(p), \quad \log_b(q^6) = 6\log_b(q), \quad \log_b(m^5) = 5\log_b(m), \quad \log_b(b^9) = 9$

Thus, the final expression is:

$2\log_b(p) + 6\log_b(q) - 5\log_b(m) - 9$

Would you like further clarification or examples?

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1. How does the base $b$ affect the simplifications of logarithms?
2. Can logarithms be expanded similarly for other bases or natural logs?
3. How would you simplify $\log_b((x^3)(y^4)/z^2)$?
4. What are the main properties of logarithms?
5. How do logarithms relate to exponents in reverse operations?

Tip: Always start by applying the quotient and product rules before simplifying powers in logarithmic expressions!