We are given the following values:

$\log a = 9, \quad \log b = -2, \quad \log c = -12$

We need to compute the logarithm of the following expression:

$\log \left( \frac{b^9}{\sqrt[3]{a^5 c^7}} \right)$

Step 1: Use logarithmic properties

We can simplify the logarithmic expression using the following properties:

1. Log of a quotient:
   $\log \left( \frac{x}{y} \right) = \log x - \log y$

2. Log of a power:
   $\log (x^n) = n \cdot \log x$

3. Log of a product:
   $\log (x \cdot y) = \log x + \log y$

Now, we apply these properties to the given expression:

[ \log \left( \frac{b^9}{\sqrt[3]{a^5 c^7}} \right) = \log (b^9) - \log \left( \sqrt[3]{a^5 c^7} \right) ]

Step 2: Simplify the first part

$\log (b^9) = 9 \log b = 9 \times (-2) = -18$

Step 3: Simplify the second part

Next, simplify the term involving the cube root:

$\log \left( \sqrt[3]{a^5 c^7} \right) = \log \left( (a^5 c^7)^{1/3} \right) = \frac{1}{3} \log (a^5 c^7)$

Now use the logarithm property of a product:

$\log (a^5 c^7) = \log (a^5) + \log (c^7) = 5 \log a + 7 \log c$

So:

$\log \left( \sqrt[3]{a^5 c^7} \right) = \frac{1}{3} \left( 5 \log a + 7 \log c \right)$

Substitute the values for $\log a$ and $\log c$:

$\log \left( \sqrt[3]{a^5 c^7} \right) = \frac{1}{3} \left( 5 \times 9 + 7 \times (-12) \right)$

$= \frac{1}{3} \left( 45 - 84 \right) = \frac{1}{3} \times (-39) = -13$

Step 4: Combine the results

Now combine the two parts:

$\log \left( \frac{b^9}{\sqrt[3]{a^5 c^7}} \right) = -18 - (-13) = -18 + 13 = -5$

Thus, the value of the logarithmic expression is:

$\boxed{-5}$

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Would you like more details or have any questions about the steps?

Here are some related questions:

1. How do logarithmic properties simplify complex expressions?
2. What is the difference between logarithms with different bases (e.g., common log vs. natural log)?
3. Can we apply the same steps for logarithms with different bases?
4. How would you calculate logarithms without a calculator using logarithmic properties?
5. What other logarithmic identities are useful in solving more complex problems?

Tip: Always remember that breaking down expressions into smaller parts and applying basic logarithmic properties step-by-step helps avoid confusion and leads to clearer solutions.