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Express the given expression as an integer or as a fraction in simplest form. log_(2)((1)/(root(5)(2))) Answer:

Express the given expression as an integer or as a fraction in simplest form.\newlinelog⁡2(125) \log _{2}\left(\frac{1}{\sqrt[5]{2}}\right) \newlineAnswer:

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Full solution

Q. Express the given expression as an integer or as a fraction in simplest form.\newlinelog⁡2(125) \log _{2}\left(\frac{1}{\sqrt[5]{2}}\right) \newlineAnswer:
  1. Apply Log Property: Apply the property of logarithms that allows the exponent on the argument of the log to be brought out as a coefficient. \newlinelog⁡2(125)=log⁡2(2−15)\log_2\left(\frac{1}{\sqrt[5]{2}}\right) = \log_2\left(2^{-\frac{1}{5}}\right)
  2. Use Log Property: Use the property of logarithms that states log⁡b(bx)=x\log_b(b^x) = x, where bb is the base of the logarithm and xx is the exponent.\newlinelog⁡2(2−1/5)=−15\log_2(2^{-1/5}) = -\frac{1}{5}
  3. Simplify Logarithm: Since the base of the logarithm and the base of the exponent are the same 22, the logarithm simplifies directly to the exponent.\newlineThe final answer is −15\frac{-1}{5}.

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