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What is the value of log_(4)root(5)(4) ? Answer:

What is the value of log⁡445 \log _{4} \sqrt[5]{4} ?\newlineAnswer:

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Full solution

Q. What is the value of log⁡445 \log _{4} \sqrt[5]{4} ?\newlineAnswer:
  1. Identify Property: Identify the property of logarithms to simplify the expression.\newlineThe fifth root of 44 can be written as 4(1/5)4^{(1/5)}. We can use the power property of logarithms, which states that log⁡b(ac)=c⋅log⁡b(a)\log_b(a^c) = c \cdot \log_b(a).
  2. Apply Power Property: Apply the power property to the logarithmic expression. \newlinelog⁡4(415)\log_{4}(4^{\frac{1}{5}}) becomes 15×log⁡4(4)\frac{1}{5} \times \log_{4}(4).
  3. Simplify Logarithm: Simplify the logarithm log⁡4(4)\log_{4}(4).\newlineSince the base of the logarithm and the number are the same, log⁡4(4)\log_{4}(4) is equal to 11.
  4. Multiply Results: Multiply the result from Step 33 by the fraction from Step 22.\newline(15)×1=15(\frac{1}{5}) \times 1 = \frac{1}{5}.
  5. Conclude Final Answer: Conclude the final answer.\newlineThe value of log⁡445\log_{4} \sqrt[5]{4} is 15\frac{1}{5}.

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