SOLUTION: solve for an exact answer 4^(x+2) = 7^(3x)

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Question 239764: solve for an exact answer 4^(x+2) = 7^(3x)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Solving equations with variables in the exponents often requires the use of logarithms. Since we are asked to find exact answers, it means that we will not use calculators to find the logarithms. We will just leave the logarithms alone. So it doesn't make any difference which base we use for the logarithms. I'll solve the problem using a couple of different bases:
  • Base 4 logarithms.
    • Find base 4 logarithm of each side:
      log%284%2C+%284%5E%28x%2B2%29%29%29+=+log%284%2C+%287%5E%283x%29%29%29
    • Use the property log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29 to move the exponents in front:
      %28x%2B2%29log%284%2C+%284%29%29+=+3x%2Alog%284%2C+%287%29%29
    • Since log%284%2C+%284%29%29+=+1, (This is why I chose base 4 logarithms)
      x%2B2+=+3x%2Alog%284%2C+%287%29%29
    • Now we solve for x. Subtract x from each side (to get the x's on one side):
      2+=+3x%2Alog%284%2C+%287%29%29+-+x
    • Factor out x on the right side:
      2+=+x%283log%284%2C+%287%29%29+-+1%29
    • Divide both sides by 3log%284%2C+%287%29%29+-+1:

      2%2F%283log%284%2C+%287%29%29+-+1%29+=+x which is an exact answer.
  • Base 7 logarithms.
    • Find base 7 logarithm of each side:
      log%287%2C+%284%5E%28x%2B2%29%29%29+=+log%287%2C+%287%5E%283x%29%29%29
    • Use the property log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29 to move the exponents in front:
      %28x%2B2%29log%287%2C+%284%29%29+=+3x%2Alog%287%2C+%287%29%29
    • Since log%287%2C+%287%29%29+=+1, (This is why I chose base 7 logarithms)
      %28x%2B2%29log%287%2C+%284%29%29+=+3x
    • Now we solve for x. Use the Distributive property on the left side:
      x%2Alog%287%2C+%284%29%29+%2B+2log%287%2C+%284%29%29+=+3x
    • Subtract x*log(7, (4)) from each side (to get the x's on one side):
      2log%287%2C+%284%29%29+=+3x+-+x%2Alog%287%2C+%284%29%29
    • Factor out x on the right side:
      2log%287%2C+%284%29%29+=+x%283+-+log%287%2C+%284%29%29%29
    • Divide both sides by 3+-+log%287%2C+%284%29%29:

      %282log%287%2C+%284%29%29%29%2F%283+-+log%287%2C+%284%29%29%29+=+x which is an exact answer.
  • Other base logarithms. These can be used. But since there will not be a log%284%2C+%284%29%29 or log%287%2C+%287%29%29, the expressions and solution will be even more complicated.