Solve the equation for \( x \). Give an exact solution if possible otherwise give an approximation to 3-decimal places. \( \log _{4}(x)-\log _{4}(x+6)=\log _{4}(x-5) \)

Solve the equation \( \log_{4}(x)-\log_{4}(x+6)=\log_{4}(x-5) \). Solve the equation by following steps: - step0: Solve for \(x\): \(\log_{4}{\left(x\right)}-\log_{4}{\left(x+6\right)}=\log_{4}{\left(x-5\right)}\) - step1: Find the domain: \(\log_{4}{\left(x\right)}-\log_{4}{\left(x+6\right)}=\log_{4}{\left(x-5\right)},x>5\) - step2: Transform the expression: \(\log_{4}{\left(\frac{x}{x+6}\right)}=\log_{4}{\left(x-5\right)}\) - step3: Set the arguments equal: \(\frac{x}{x+6}=x-5\) - step4: Multiply both sides of the equation by LCD: \(\frac{x}{x+6}\times \left(x+6\right)=\left(x-5\right)\left(x+6\right)\) - step5: Simplify the equation: \(x=x^{2}+x-30\) - step6: Cancel equal terms: \(0=x^{2}-30\) - step7: Swap the sides: \(x^{2}-30=0\) - step8: Move the constant to the right side: \(x^{2}=30\) - step9: Simplify the expression: \(x=\pm \sqrt{30}\) - step10: Separate into possible cases: \(\begin{align}&x=\sqrt{30}\\&x=-\sqrt{30}\end{align}\) - step11: Check if the solution is in the defined range: \(\begin{align}&x=\sqrt{30}\\&x=-\sqrt{30}\end{align},x>5\) - step12: Find the intersection: \(x=\sqrt{30}\) The exact solution to the equation \( \log_{4}(x)-\log_{4}(x+6)=\log_{4}(x-5) \) is \( x = \sqrt{30} \). If you need an approximation to 3-decimal places, the solution is \( x \approx 5.477 \).