Step 1 :The given equation is: \(\ln (8)+\ln (8 x^{2}-4)=\ln (137)\)
Step 2 :Using the property of logarithms that the sum of the logarithms of two numbers is equal to the logarithm of the product of those two numbers, we can rewrite the left side of the equation as: \(\ln (8(8x^{2}-4))=\ln (137)\)
Step 3 :Simplifying the left side gives: \(\ln (64x^{2}-32)=\ln (137)\)
Step 4 :Since the logarithms on both sides of the equation are equal, we can set their arguments equal to each other: \(64x^{2}-32=137\)
Step 5 :Solving this equation for x gives: \(64x^{2}=137+32\)
Step 6 :\(64x^{2}=169\)
Step 7 :\(x^{2}=\frac{169}{64}\)
Step 8 :Taking the square root of both sides gives two possible solutions: \(x=\pm\sqrt{\frac{169}{64}}\)
Step 9 :Simplifying the square root gives: \(x=\pm\frac{13}{8}\)
Step 10 :Therefore, the solutions to the equation are \(\boxed{x = \frac{13}{8}}\) and \(\boxed{x = -\frac{13}{8}}\)