Start with the equation \(\ln(x+5) = 1 + \ln x\).

We know that \(1 = \ln e\), so the equation becomes \(\ln(x+5) = \ln e + \ln x\).

Using the logarithm property \(\ln a + \ln b = \ln(ab)\), we can rewrite the equation as \(\ln(x+5) = \ln(ex)\).

Since the logarithms are equal, the arguments must be equal: \(x+5 = ex\).

Rearrange to solve for \(x\):

\(x + 5 = ex\)

\(5 = ex - x\)

\(5 = x(e - 1)\)

\(x = \frac{5}{e-1}\)