Having wrapped up your work on the smaller projects, you are finally able to turn your attention to the project you are most excited about research! Being one of Casey's projects, the files needed are password protected. Once again, you review the hint that Casey has left as a reminder of the password. Find an exact solution for
\[
\ln (8)+\ln \left(8 x^{2}-4\right)=\ln (137)
\]

If there is no solution, enter NA. If there are multiple solutions, they should be separated by semicolons (e.g. $2 ; 4 ; 6$ or $x+1 ; x-1$ ).
\[
x=\square \text { 중 }
\]

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 \(\boxed{x = -\frac{13}{8}}\)