SOLUTION: Simplify the logarithm log1/3 3 root 81

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Question 431345: Simplify the logarithm
log1/3 3 root 81
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
I think that the reason you haven't gotten help before now is that we don't know if the expression is
log%281%2F3%2C+%283%2Asqrt%2881%29%29%29
or
log%281%2F3%2C+%28root%283%2C+81%29%29%29
or something else. The word "root" is generic. It applies to all kinds of roots: square (or 2nd), cube (or 3rd), 4th, 5th, etc. So please specify the specific type of root. So if the first expression is correct say "square root" or "second root". If the second expression is correct then say "cube root" or "3rd root". (Or click on the "Show source" link just above this reply and see what I typed to get these expressions to display so nicely.)

Without knowing what your problem is I cannot give you specific help. But I can give you some clues that may help regardless of which expression is correct:
  • The 1/3 and the 81 (and, in the first expression, the 3) are all powers of 3: 1%2F3+=+3%5E%28-1%29, 81+=+3%5E4 and 3+=+3%5E1. So rewrite the 81 as 3%5E4 and use the change of base formula, log%28a%2C%28p%29%29+=+log%28b%2C+%28p%29%29%2Flog%28b%2C+%28q%29%29, to convert the base 1/3 log into a fraction of base 3 logarithms.
  • Roots can be expressed as fractional exponents. Square roots are exponents of 1/2 and cube roots are exponents of 1/3. Rewrite the "root" of 3%5E4 as 3%5E4 to the 1/2 or 1/3 power depending on what type of root it is.
  • Use the rules for exponents to simplify the argument down to 3 to some power.
  • Use one of the properties of logarithms, log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29, to move the exponent on the 3 out in front.
  • You sould now have an expression involving base 3 logarithms whose values you (should) know. The rest should be obvious.