Use the one-to-one property of logarithms to find an exact solution for
\[
\ln (7)+\ln \left(7 x^{2}-6\right)=\ln (127)
\]
If there is no solution, enter NA.
The field below accepts a list of numbers or formulas separated by semicolons (e.g. $2 ; 4 ; 6$ or $x+1 ; x-1)$.
\[
x=
\]
Answer
So, the final answer is \(x = \boxed{\frac{13}{7}}\).
Step 1 ：Use the one-to-one property of logarithms to set the two sides of the equation equal to each other: \(7(7x^2 - 6) = 127\).
Step 2 ：Simplify the equation to get: \(49x^2 - 42 = 127\).
Step 3 ：Solve the equation to get two solutions: \(x = -\frac{13}{7}\) and \(x = \frac{13}{7}\).
Step 4 ：Since the domain of the logarithm function is \((0, \infty)\), we can't have a negative argument for the logarithm. Therefore, the only valid solution is \(x = \frac{13}{7}\).
Step 5 ：So, the final answer is \(x = \boxed{\frac{13}{7}}\).
