Step 1 :\(\ln (5)+\ln (5x^{2}-2)=\ln (159)\)
Step 2 :Using the property of logarithms, \(\ln (5(5x^2 - 2)) = \ln (159)\)
Step 3 :Equating the arguments, \(5(5x^2 - 2) = 159\)
Step 4 :Solving for x, \(5x^2 - 2 = 159/5\)
Step 5 :\(5x^2 = 159/5 + 2\)
Step 6 :\(5x^2 = 31.8 + 2\)
Step 7 :\(5x^2 = 33.8\)
Step 8 :\(x^2 = 33.8/5\)
Step 9 :\(x^2 = 6.76\)
Step 10 :Taking the square root of both sides, \(x = \sqrt{6.76}\) or \(x = -\sqrt{6.76}\)
Step 11 :Checking the solutions in the original equation, for x = 2.6: \(\ln (5)+\ln (5 (2.6)^{2}-2)=\ln (159)\)
Step 12 :This is a true statement, so x = 2.6 is a valid solution.
Step 13 :Checking the solutions in the original equation, for x = -2.6: \(\ln (5)+\ln (5 (-2.6)^{2}-2)=\ln (159)\)
Step 14 :This is also a true statement, so x = -2.6 is a valid solution.
Step 15 :\(\boxed{x = 2.6, x = -2.6}\)