SOLUTION: log(base 5)(x-2) + log (base 5)(x-6) =1. Find all of the values of x.

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Question 1153079: log(base 5)(x-2) + log (base 5)(x-6) =1. Find all of the values of x.
Answer by ikleyn(52866) About Me  (Show Source):
You can put this solution on YOUR website!
.

From the given equation,


    (x-2)*(x-6) = 5.    (1)


The numbers, whose product is 5 and that differ 4 units, are the pairs  (1,5) and (-1,-5).


So,  x-2 = 5,  x = 5 + 2 = 7  is one solution,  while

     x-2 = -1,  x = 1   is formally another solution to the quadratic equation (1).


But under the logarithm, only the solution x = 7 survives.


ANSWER.  x = 7 is the ONLY solution to the given logarithmic equation.

Solved.