Use the one-to-one property of logarithms to solve.
\[
\ln \left(x^{2}-2\right)+\ln (9)=\ln (7)
\]
Enter the exact answers.
The field below accepts a list of numbers or formulas separated by semicolons (e.g. $2 ; 4 ; 6$ or $x+1 ; x-1)$
\[
x=
\]
Show your work and explain, in your own words, how you arrived at your answer.
\(\boxed{x=\frac{5}{3}, x=-\frac{5}{3}}\)
Step 1 :\(\ln ((x^{2}-2) \cdot 9)=\ln (7)\)
Step 2 :\(\ln (9x^{2}-18)=\ln (7)\)
Step 3 :\(9x^{2}-18=7\)
Step 4 :\(9x^{2}=25\)
Step 5 :\(x^{2}=\frac{25}{9}\)
Step 6 :\(x=\pm \sqrt{\frac{25}{9}}\)
Step 7 :\(x=\pm \frac{5}{3}\)
Step 8 :Check the solutions: For \(x=\frac{5}{3}\), \(x^{2}-2=\left(\frac{5}{3}\right)^{2}-2=\frac{7}{9}\), which is positive. For \(x=-\frac{5}{3}\), \(x^{2}-2=\left(-\frac{5}{3}\right)^{2}-2=\frac{7}{9}\), which is also positive.
Step 9 :\(\boxed{x=\frac{5}{3}, x=-\frac{5}{3}}\)