Q5: Solve the equation: \( \log _{6} x+\log _{6}(x+5)=2 \). Be sure to check your possible solutions.

Solve the equation \( \log_{6}x+\log_{6}(x+5)=2 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\log_{6}{\left(x\right)}+\log_{6}{\left(x+5\right)}=2\) - step1: Find the domain: \(\log_{6}{\left(x\right)}+\log_{6}{\left(x+5\right)}=2,x>0\) - step2: Transform the expression: \(\log_{6}{\left(x\left(x+5\right)\right)}=2\) - step3: Convert the logarithm into exponential form: \(x\left(x+5\right)=6^{2}\) - step4: Evaluate the power: \(x\left(x+5\right)=36\) - step5: Expand the expression: \(x^{2}+5x=36\) - step6: Move the expression to the left side: \(x^{2}+5x-36=0\) - step7: Factor the expression: \(\left(x-4\right)\left(x+9\right)=0\) - step8: Separate into possible cases: \(\begin{align}&x-4=0\\&x+9=0\end{align}\) - step9: Solve the equation: \(\begin{align}&x=4\\&x=-9\end{align}\) - step10: Check if the solution is in the defined range: \(\begin{align}&x=4\\&x=-9\end{align},x>0\) - step11: Find the intersection: \(x=4\) The solution to the equation \( \log _{6} x+\log _{6}(x+5)=2 \) is \( x=4 \). To check the solution, we substitute \( x=4 \) back into the original equation: \( \log _{6} 4+\log _{6}(4+5)=2 \) \( \log _{6} 4+\log _{6} 9=2 \) \( \log _{6} 36=2 \) Since \( \log _{6} 36 = 2 \) is true, the solution \( x=4 \) is correct.