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Express the given expression as an integer or as a fraction in simplest form. log_(8)((1)/(8^(7))) Answer:

Express the given expression as an integer or as a fraction in simplest form.\newlinelog⁡8(187) \log _{8}\left(\frac{1}{8^{7}}\right) \newlineAnswer:

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Full solution

Q. Express the given expression as an integer or as a fraction in simplest form.\newlinelog⁡8(187) \log _{8}\left(\frac{1}{8^{7}}\right) \newlineAnswer:
  1. Understand Logarithm Expression: Understand the logarithm expression.\newlineThe expression log⁡8(187)\log_{8}\left(\frac{1}{8^{7}}\right) asks for the power to which the base 88 must be raised to obtain the value 187\frac{1}{8^{7}}.
  2. Use Logarithm Power Rule: Use the logarithm power rule.\newlineThe power rule of logarithms states that log⁡a(bc)=c⋅log⁡a(b)\log_{a}(b^{c}) = c \cdot \log_{a}(b). In this case, we can apply the inverse of this rule because we have a fraction inside the logarithm, which can be written as a negative exponent: log⁡8(8−7)\log_{8}(8^{-7}).
  3. Simplify Logarithm: Simplify the logarithm.\newlineSince the base of the logarithm and the base of the exponent are the same 88, the logarithm simplifies to the exponent: log⁡8(8−7)=−7\log_{8}(8^{-7}) = -7.
  4. Verify Result: Verify the result.\newlineWe can check our result by converting it back to exponential form: 8−7=1878^{-7} = \frac{1}{8^{7}}, which confirms that our logarithm simplification is correct.

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