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log_(25)(1)/(5)=

log⁡2515= \log _{25} \frac{1}{5}=

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Full solution

Q. log⁡2515= \log _{25} \frac{1}{5}=
  1. Recognize base and argument: Recognize the base of the logarithm and the argument.\newlineThe base of the logarithm is 2525, and the argument is 15\frac{1}{5}. We need to find the exponent to which 2525 must be raised to get 15\frac{1}{5}.
  2. Express 15\frac{1}{5} as a power: Express 15\frac{1}{5} as a power of 2525.\newlineSince 2525 is 55 squared (525^2), we can express 15\frac{1}{5} as 5−15^{-1}, which is the same as (52)−12(5^2)^{-\frac{1}{2}}.
  3. Rewrite using new expression: Rewrite the logarithm using the new expression for 15\frac{1}{5}. \newlinelog⁡25(15)\log_{25}\left(\frac{1}{5}\right) becomes log⁡25((52)−12)\log_{25}\left((5^2)^{-\frac{1}{2}}\right).
  4. Apply power property: Apply the power property of logarithms.\newlineThe power property of logarithms states that log⁡b(ac)=c⋅log⁡b(a)\log_b(a^c) = c \cdot \log_b(a). Therefore, log⁡25((52)−12)\log_{25}((5^2)^{-\frac{1}{2}}) becomes (−12)⋅log⁡25(52)(-\frac{1}{2}) \cdot \log_{25}(5^2).
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