SOLUTION: Given that {{{3^(x+2)}}} = {{{6^(2x)}}}. Show that x = {{{log(12,(9))}}}

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Question 533803: Given that 3%5E%28x%2B2%29 = 6%5E%282x%29. Show that x = log%2812%2C%289%29%29
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
                   3%5E%28x%2B2%29 = 6%5E%282x%29

Take log base 12 of both sides:

              log%2812%2C%283%5E%28x%2B2%29%29%29 = log%2812%2C%286%5E%282x%29%29%29

              %28x%2B2%29log%2812%2C%283%29%29 = 2x%2Alog%2812%2C%286%29%29

              %28x%2B2%29log%2812%2C%283%29%29 = 2x%2Alog%2812%2C%286%29%29

              %28x%2B2%29log%2812%2C%283%29%29 = 2x%2Alog%2812%2C%283%2A2%29%29

              %28x%2B2%29log%2812%2C%283%29%29 = 2x%28log%2812%2C%283%29%29%2Blog%2812%2C%282%29%29%29

To make things easier, let log%2812%2C%283%29%29 = A and let log%2812%2C%282%29%29 = B

              (x + 2)*A = 2x(A + B)
                Ax + 2A = 2Ax + 2Bx 
                     2A = Ax + 2Bx
                     2A = x(A + 2B)
                  %282A%29%2F%28A%2B2B%29 = x
Therefore:

x = %282A%29%2F%28A%2B2B%29 = %282log%2812%2C%283%29%29%29%2F%28log%2812%2C%283%29%29%2B2log%2812%2C%282%29%29%29 = %28log%2812%2C%283%5E2%29%29%29%2F%28log%2812%2C%283%29%29%2Blog%2812%2C%282%5E2%29%29%29 = %28log%2812%2C%289%29%29%29%2F%28log%2812%2C%283%29%29%2Blog%2812%2C%284%29%29%29 = %28log%2812%2C%289%29%29%29%2F%28log%2812%2C%283%2A4%29%29%29 = %28log%2812%2C%289%29%29%29%2F%28log%2812%2C%2812%29%29%29 = %28log%2812%2C%289%29%29%29%2F1 = log%2812%2C%289%29%29
               
Edwin