To simplify the expression [tex]\(\log(9x^5) + 5 \log\left(\frac{1}{x}\right)\)[/tex], let's use the properties of logarithms step-by-step.
First, recall the logarithm properties:
1. [tex]\(\log(ab) = \log a + \log b\)[/tex]
2. [tex]\(\log\left(\frac{a}{b}\right) = \log a - \log b\)[/tex]
3. [tex]\(\log(a^b) = b \log a\)[/tex]
Given expression: [tex]\(\log(9x^5) + 5 \log\left(\frac{1}{x}\right)\)[/tex]
We will handle each part of the expression individually:
### Step 1: Simplify [tex]\(\log(9x^5)\)[/tex]
Using the property [tex]\(\log(ab) = \log a + \log b\)[/tex]:
[tex]\[
\log(9x^5) = \log 9 + \log x^5
\][/tex]
Then, using the property [tex]\(\log(a^b) = b \log a\)[/tex]:
[tex]\[
\log x^5 = 5 \log x
\][/tex]
So,
[tex]\[
\log(9x^5) = \log 9 + 5 \log x
\][/tex]
### Step 2: Simplify [tex]\(5 \log\left(\frac{1}{x}\right)\)[/tex]
Using the property [tex]\(\log \left(\frac{a}{b}\right) = \log a - \log b\)[/tex]:
[tex]\[
\log\left(\frac{1}{x}\right) = \log 1 - \log x
\][/tex]
Since [tex]\(\log 1 = 0\)[/tex],
[tex]\[
\log\left(\frac{1}{x}\right) = -\log x
\][/tex]
Then multiply by 5:
[tex]\[
5 \log\left(\frac{1}{x}\right) = 5(-\log x) = -5 \log x
\][/tex]
### Step 3: Combine the simplified parts
Now, combining the simplified parts from Step 1 and Step 2:
[tex]\[
\log(9x^5) + 5 \log\left(\frac{1}{x}\right) = (\log 9 + 5 \log x) + (-5 \log x)
\][/tex]
Observe that [tex]\(5 \log x - 5 \log x\)[/tex] cancels out:
[tex]\[
\log(9x^5) + 5 \log\left(\frac{1}{x}\right) = \log 9
\][/tex]
### Conclusion:
Thus, the simplified form of [tex]\(\log(9x^5) + 5 \log\left(\frac{1}{x}\right)\)[/tex] is [tex]\(\log 9\)[/tex].
Hence, the correct answer is:
A. [tex]\(\log 9\)[/tex]
