Consider the Equation $\log _{6} x+\log _{6}(x-5)=1$
(A) Solve the Equation Analytically

Essay

Consider the equation $\log _{6} x+\log _{6}(x-5)=1$ .
(a) Solve the equation analytically. If there is an extraneous value, what is it?
(b) To support the solution in part (a), we may graph $y_{1}=\log _{6} x+\log _{6}(x-5)-1$ and find the $x$ -intercept.
Write an expression for $y_{1}$ using the change-of-base rule with base 10, and graph the function to support the solution from part (a).
(c) Use the graph to solve the inequality $\log _{6} x+\log _{6}(x-5)<1$ .