Start with the equation \(\log_{10}(x+9) = 2 + \log_{10} x\).

Use the logarithm property: \(\log_{10}(x+9) = \log_{10}(10^2) + \log_{10} x\).

This simplifies to \(\log_{10}(x+9) = \log_{10}(100x)\).

Since the logarithms are equal, set the arguments equal: \(x+9 = 100x\).

Rearrange to solve for \(x\):

\(x + 9 = 100x\)

\(9 = 99x\)

\(x = \frac{9}{99} = \frac{1}{11}\).