Please explain and show work pls help

Question

Consider the equation log((x + 3)(x−8))
+ log(x+3/x-8) = 2.

 

(a) Explain why x = 7 is not a solution to this equation.

 

(b) By combining the logarithms, when appropriate, solve the equation.

Answer 1

(a).  Substituting 7 into the equation yields log(-10) + log(-10) = 2.              Since the logarithm of a negative number is undefined, 7 is not a          solution of the equation.

 

(b).  log((x+3)(x-8)) + log((x+3)/(x-8)) = 2

 

       log[(x+3)(x-8)(x+3)/(x-8)] = 2

 

       log(x+3)² = 2

 

       Converting to exponential form, we get (x+3)² = 10² = 100

 

       So, x+3 = ±10

 

       x = 7 or x = -13

 

       We know from (a) that 7 is not a solution.  Checking -13 in the            original equation shows that -13 is a solution.

 

        The only solution is -13.