SOLUTION: log 8 (sqrt 8^9) = n

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Question 240807: log 8 (sqrt 8^9) = n
Found 2 solutions by jsmallt9, JimboP1977:
Answer by jsmallt9(3759) About Me  (Show Source):
You can put this solution on YOUR website!
log%288%2C+%28sqrt%288%5E9%29%29%29+=+n
As usual, there are several ways to solve this. Maybe the quickest way is based on
  • knowing our exponents (including fractional exponents) well
  • seeing that the argument of the base 8 logarithm is a power of 8
  • knowing that you do not need a calculator to find the base 8 logarithm of a power of 8

Since square roots are the same as raising to the 1/2 power we can rewrite the argument:
log%288%2C+%28%288%5E9%29%5E%281%2F2%29%29%29+=+n
Using the property of exponents, %28a%5Ep%29%5Eq+=+a%5E%28p%2Aq%29, we get:
log%288%2C+%288%5E%289%2F2%29%29%29+=+n
Using the property of logarithms, log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29, we can move the exponent out in front:
%289%2F2%29log%288%2C+%288%29%29+=+n
Since by definition log%28a%2C+%28a%29%29+=+1:
9%2F2+=+n

Answer by JimboP1977(311) About Me  (Show Source):
You can put this solution on YOUR website!
Log(base 8) (8^9)^(1/2) = n
Log(base 8) (8^4.5) = n
4.5 * Log(base 8) (8) = n
4.5 = n