Find the numerical value of the log expression.
log, a, equals, 9, log, b, equals, minus, 2, log, c, equals, minus, 12
loga=9logb=−2logc=−12
log, start fraction, b, to the power 9 , divided by, cube root of, a, to the power 5 , c, to the power 7 , end cube root, end fraction
log 
3
  
a 
5
 c 
7
 
​
 
b 
9
 
​
 
Answer


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:

\[
\boxed{-5}
\]

---

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.

log a = 9, log b = -2, log c = -12. Find the numerical value of log (b^9 / cube root of (a^5 c^7)).

Learn how to simplify logarithmic expressions with powers and cube roots. This problem uses logarithmic properties to solve an expression involving b^9 divided by the cube root of a^5 c^7. Key concepts include log of a quotient, log of a power, and log of a product. Suitable for students in Grades 10-12.

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