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log_(3)(1)/(27)=

log⁡3127= \log _{3} \frac{1}{27}=

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Full solution

Q. log⁡3127= \log _{3} \frac{1}{27}=
  1. Recognize Power of 33: We are asked to find the value of the logarithm of 127\frac{1}{27} with base 33. The first step is to recognize that 2727 is a power of 33. \newline27=3327 = 3^3
  2. Rewrite as Negative Power: Now, we can rewrite 127\frac{1}{27} as 33 raised to a negative power because 127\frac{1}{27} is the reciprocal of 2727. \newline127=133=3−3\frac{1}{27} = \frac{1}{3^3} = 3^{-3}
  3. Apply Logarithm Definition: Next, we can apply the definition of a logarithm. The logarithm log⁡3(3−3)\log_{3}(3^{-3}) asks for the exponent that the base 33 must be raised to in order to yield 3−33^{-3}.\newlinelog⁡3(3−3)=−3\log_{3}(3^{-3}) = -3
  4. Final Logarithm Value: Since the base 33 raised to the power of −3-3 gives us 127\frac{1}{27}, the logarithm log⁡3(127)\log_{3}(\frac{1}{27}) is equal to −3-3.

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