How do you solve \( \frac{1}{2} \log _{a}(x+2)+\frac{1}{2} \log _{a}(x-1)=\frac{2}{3} \log _{a} 27 \)

Solve the equation \( \frac{1}{2} \log _{a}(x+2)+\frac{1}{2} \log _{a}(x-1)=\frac{2}{3} \log _{a} 27 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{1}{2}\log_{a}{\left(x+2\right)}+\frac{1}{2}\log_{a}{\left(x-1\right)}=\frac{2}{3}\log_{a}{\left(27\right)}\) - step1: Add the terms: \(\frac{1}{2}\log_{a}{\left(\left(x+2\right)\left(x-1\right)\right)}=\frac{2}{3}\log_{a}{\left(27\right)}\) - step2: Multiply by the reciprocal: \(\frac{1}{2}\log_{a}{\left(\left(x+2\right)\left(x-1\right)\right)}\times 2=\frac{2}{3}\log_{a}{\left(27\right)}\times 2\) - step3: Multiply: \(\log_{a}{\left(\left(x+2\right)\left(x-1\right)\right)}=\frac{4}{3}\log_{a}{\left(27\right)}\) - step4: Evaluate the logarithm: \(\left(x+2\right)\left(x-1\right)=27^{\frac{4}{3}}\) - step5: Expand the expression: \(x^{2}+x-2=27^{\frac{4}{3}}\) - step6: Expand the expression: \(x^{2}+x-2=81\) - step7: Move the expression to the left side: \(x^{2}+x-2-81=0\) - step8: Subtract the numbers: \(x^{2}+x-83=0\) - step9: Solve using the quadratic formula: \(x=\frac{-1\pm \sqrt{1^{2}-4\left(-83\right)}}{2}\) - step10: Simplify the expression: \(x=\frac{-1\pm \sqrt{333}}{2}\) - step11: Simplify the expression: \(x=\frac{-1\pm 3\sqrt{37}}{2}\) - step12: Separate into possible cases: \(\begin{align}&x=\frac{-1+3\sqrt{37}}{2}\\&x=\frac{-1-3\sqrt{37}}{2}\end{align}\) - step13: Rewrite the fraction: \(\begin{align}&x=\frac{-1+3\sqrt{37}}{2}\\&x=-\frac{1+3\sqrt{37}}{2}\end{align}\) - step14: Check if the solution is in the defined range: \(\begin{align}&x=\frac{-1+3\sqrt{37}}{2}\\&x=-\frac{1+3\sqrt{37}}{2}\end{align},x>1\) - step15: Find the intersection: \(x=\frac{-1+3\sqrt{37}}{2}\) The solution to the equation \( \frac{1}{2} \log _{a}(x+2)+\frac{1}{2} \log _{a}(x-1)=\frac{2}{3} \log _{a} 27 \) is \( x = \frac{-1+3\sqrt{37}}{2} \).