We are solving the equation:Nbsp;
\[4 \log_{10} x + 4 \log_{100} x + 8 \log_{1000} x = 13.\]
Step 1: Rewrite logs in base 10.
Using logarithm properties:
\[\log_{100} x = \frac{\log_{10} x}{\log_{10} 100} = \frac{\log_{10} x}{2} = \frac{1}{2} \log_{10} x,\]
\[\log_{1000} x = \frac{\log_{10} x}{\log_{10} 1000} = \frac{\log_{10} x}{3} = \frac{1}{3} \log_{10} x.\]
Substitute these into the original equation:
\[4 \log_{10} x + 4 \times \frac{1}{2} \log_{10} x + 8 \times \frac{1}{3} \log_{10} x = 13.\]
Step 2: Simplify the terms.
\[4 \log_{10} x + 2 \log_{10} x + \frac{8}{3} \log_{10} x = 13.\]
Combine the coefficients:
\[\left( 4 + 2 + \frac{8}{3} \right) \log_{10} x = 13.\]
Sum the coefficients:
\[4 + 2 + \frac{8}{3} = \frac{12}{3} + \frac{6}{3} + \frac{8}{3} = \frac{26}{3}.\]
This gives:
\[\frac{26}{3} \log_{10} x = 13.\]
Step 3: Solve for \(\log_{10} x\).
Multiply by \(\frac{3}{26}\):
\[\log_{10} x = 13 \times \frac{3}{26} = \frac{39}{26} = \frac{3}{2}.\]
Step 4: Solve for \(x\).
Convert \(\log_{10} x = \frac{3}{2}\) to exponential form:
\[x = 10^{\frac{3}{2}} = \sqrt{10^3} = \sqrt{1000}.\]
Simplify:
\[x = 31.622.\]
Step 5: Find the greatest integer not exceeding \(x\).
The greatest integer not exceeding \(x = 31.622\) is:
\[\boxed{31}.\]