To solve the expression:

$\log \frac{b^7}{a^3 c^2}$

given that: $\log a = 4, \quad \log b = 9, \quad \log c = 4,$

we can proceed as follows:

Step 1: Apply the Logarithm Quotient Rule

Using the rule $\log \frac{X}{Y} = \log X - \log Y$, we can rewrite the expression as: $\log \frac{b^7}{a^3 c^2} = \log b^7 - \log(a^3 c^2)$

Step 2: Apply the Logarithm Power Rule

Using the rule $\log X^n = n \cdot \log X$, we get: $\log b^7 = 7 \cdot \log b \quad \text{and} \quad \log(a^3 c^2) = \log a^3 + \log c^2 = 3 \cdot \log a + 2 \cdot \log c$

Substituting these into the expression, we get: $\log \frac{b^7}{a^3 c^2} = 7 \cdot \log b - (3 \cdot \log a + 2 \cdot \log c)$

Step 3: Substitute the Given Values

Now, we substitute $\log a = 4$, $\log b = 9$, and $\log c = 4$: $= 7 \cdot 9 - (3 \cdot 4 + 2 \cdot 4)$

Calculating each part:

1. $7 \cdot 9 = 63$
2. $3 \cdot 4 = 12$
3. $2 \cdot 4 = 8$

Thus, $= 63 - (12 + 8) = 63 - 20 = 43$

Final Answer

$\log \frac{b^7}{a^3 c^2} = 43$

Would you like further clarification on any part of this solution?

Here are some related questions to explore:

1. How would the solution change if $\log b$ had a different value?
2. Can you apply similar techniques to simplify $\log \frac{a^5 c^4}{b^2}$?
3. What are the general logarithmic rules for products, quotients, and powers?
4. How does changing the base of the logarithm affect the solution?
5. How would you simplify expressions with different bases, like $\log_{10}$ and $\ln$?

Tip: Remember that logarithmic rules for exponents are essential when simplifying expressions, especially when dealing with products or quotients.