Certainly! Let's systematically break down and solve the given equation step-by-step.
Given equation:
[tex]\[
\log_2(x) \cdot \log_x\left(\frac{\log_{|x|} \sqrt[9]{3}}{\log_{\sqrt{3}} |x|}\right) = -1
\][/tex]
First, recognize the logarithmic expressions and simplify them using properties of logarithms.
1. Simplify [tex]\(\log_{|x|} \sqrt[9]{3}\)[/tex]:
[tex]\[
\log_{|x|} \sqrt[9]{3} = \log_{|x|}(3^{1/9})
\][/tex]
Using the logarithmic property [tex]\(\log_b(a^c) = c \log_b(a)\)[/tex], we can write:
[tex]\[
\log_{|x|} (3^{1/9}) = \frac{1}{9} \log_{|x|} (3)
\][/tex]
2. Simplify [tex]\(\log_{\sqrt{3}} |x|\)[/tex]:
[tex]\[
\log_{\sqrt{3}} |x| = \frac{\log |x|}{\log \sqrt{3}}
\][/tex]
Since [tex]\(\sqrt{3} = 3^{1/2}\)[/tex], we can write:
[tex]\[
\log_{\sqrt{3}} |x| = \frac{\log |x|}{\frac{1}{2} \log 3} = \frac{2 \log |x|}{\log 3}
\][/tex]
3. Combine these results into the given equation:
Substitute the simplified results into the equation:
[tex]\[
\log_2(x) \cdot \log_x\left(\frac{\frac{1}{9} \log_{|x|} (3)}{\frac{2 \log |x|}{\log 3}}\right) = -1
\][/tex]
Simplify the fraction inside the logarithm:
[tex]\[
\frac{\frac{1}{9} \log_{|x|} (3)}{\frac{2 \log |x|}{\log 3}} = \frac{\log_{|x|} (3)}{18 \log |x| / \log 3}
= \frac{(\log 3) / (\log |x|)}{18 \log |x| / \log 3}
= \frac{\log 3}{18 \log |x|}
\][/tex]
Thus,
[tex]\[
\log_2(x) \cdot \log_x\left(\frac{\log 3}{18 \log |x|}\right) = -1
\][/tex]
4. Simplifying the logarithm inside further:
Use the property [tex]\(\log_x(a/b) = \log_x(a) - \log_x(b)\)[/tex]:
[tex]\[
\log_2(x) \cdot \left(\log_x(\log 3) - \log_x(18 \log |x|)\right) = -1
\][/tex]
5. Expanding [tex]\(\log_x(18 \log |x|)\)[/tex]:
[tex]\[
\log_x(18 \log |x|) = \log_x(18) + \log_x(\log |x|)
= \frac{\log 18}{\log x} + \log_x(\log |x|)
\][/tex]
Combine terms:
\[
\log_2(x) \cdot \left(\frac{\log (\log 3) -\left(\log 18 + \log (\log |x|)\right) }{\log(x)}\right) = -1
= -\frac{\log (\log 3) -\left(\log 18 + \log (\log |x|)\right )}{\log(x)} = \frac{1}{x^2}
}
We can expect \(\boxed{{3}
