Start with the equation:

\(2 \ln(x + 4) - \ln x = \ln(x + a)\)

Apply the logarithm property \(a \ln b = \ln(b^a)\):

\(\ln((x + 4)^2) - \ln x = \ln(x + a)\)

Use the property \(\ln a - \ln b = \ln\left(\frac{a}{b}\right)\):

\(\ln\left(\frac{(x + 4)^2}{x}\right) = \ln(x + a)\)

Since the logarithms are equal, set the arguments equal:

\(\frac{(x + 4)^2}{x} = x + a\)

Clear the fraction by multiplying through by \(x\):

\((x + 4)^2 = x(x + a)\)

Expand both sides:

\(x^2 + 8x + 16 = x^2 + ax\)

Subtract \(x^2\) from both sides:

\(8x + 16 = ax\)

Rearrange to solve for \(x\):

\(ax - 8x = 16\)

\(x(a - 8) = 16\)

Divide by \(a - 8\):

\(x = \frac{16}{a - 8}\)