Question

if \\(\ln x + \ln(x - 6) = \ln(5x)\\), then \\(x = \\)
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Explanation:

Step1: Use logarithm property

According to the property \(\ln a+\ln b = \ln(ab)\), the left - hand side of the equation \(\ln x+\ln(x - 6)=\ln(5x)\) can be rewritten as \(\ln[x(x - 6)]\). So the equation becomes \(\ln[x(x - 6)]=\ln(5x)\).

Step2: Remove the logarithm

Since \(\ln A=\ln B\) implies \(A = B\) (for \(A>0,B>0\)), we have \(x(x - 6)=5x\).
Expand the left - hand side: \(x^{2}-6x = 5x\).

Step3: Rearrange the equation

Move all terms to one side: \(x^{2}-6x-5x=0\), which simplifies to \(x^{2}-11x = 0\).
Factor out \(x\): \(x(x - 11)=0\).
So \(x = 0\) or \(x=11\).

Step4: Check the domain of the original logarithmic equation

For \(\ln x\), \(x>0\); for \(\ln(x - 6)\), \(x-6>0\) (i.e., \(x>6\)); for \(\ln(5x)\), \(5x>0\) (i.e., \(x>0\)).
When \(x = 0\), \(\ln x\) and \(\ln(x - 6)\) are not defined.
When \(x=11\), \(\ln(11)\), \(\ln(11 - 6)=\ln(5)\) and \(\ln(5\times11)=\ln(55)\) are all defined.

Answer:

\(11\)