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What is the value of log_(6)root(5)(216) ? Answer:

What is the value of log⁡62165 \log _{6} \sqrt[5]{216} ?\newlineAnswer:

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Full solution

Q. What is the value of log⁡62165 \log _{6} \sqrt[5]{216} ?\newlineAnswer:
  1. Identify Value and Root: Identify the value inside the logarithm and the type of root.\newlineThe fifth root of 216216 is being taken, which can be written as 21615216^{\frac{1}{5}}.
  2. Calculate Fifth Root: Calculate the fifth root of 216216.\newlineSince 216=63216 = 6^3, the fifth root of 216216 is 6356^{\frac{3}{5}}.
  3. Apply Logarithm: Apply the logarithm to the result from Step 22.\newlineWe need to find the value of log⁡6(635)\log_6(6^{\frac{3}{5}}).
  4. Use Logarithm Property: Use the property of logarithms that log⁡b(bx)=x\log_b(b^x) = x.\newlineApplying this property, we get log⁡6(635)=35\log_6(6^{\frac{3}{5}}) = \frac{3}{5}.

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