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

Identify Value and Root: Identify the value inside the logarithm and the type of root. The fifth root of 216 is being taken, which can be written as $216^{\frac{1}{5}}$.Calculate Fifth Root: Calculate the fifth root of 216. Since $216 = 6^3$, the fifth root of 216 is $6^{\frac{3}{5}}$.Apply Logarithm: Apply the logarithm to the result from Step 2. We need to find the value of $\log_6(6^{\frac{3}{5}})$.Use Logarithm Property: Use the property of logarithms that $\log_b(b^x) = x$. Applying this property, we get $\log_6(6^{\frac{3}{5}}) = \frac{3}{5}$.

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

Evaluate. $\newline$ $\log _{6} \sqrt[5]{216}$ $\newline$ Write your answer in simplest form.

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Math Problems

Precalculus

Product property of logarithms

Full solution

Q. Evaluate.

\newline

\log _{6} \sqrt[5]{216}

\newline

Write your answer in simplest form.

Identify Value and Root: Identify the value inside the logarithm and the type of root. $\newline$ The fifth root of $216$ is being taken, which can be written as $216^{\frac{1}{5}}$ .
Calculate Fifth Root: Calculate the fifth root of $216$ . $\newline$ Since $216 = 6^3$ , the fifth root of $216$ is $6^{\frac{3}{5}}$ .
Apply Logarithm: Apply the logarithm to the result from Step $2$ . $\newline$ We need to find the value of $\log_6(6^{\frac{3}{5}})$ .
Use Logarithm Property: Use the property of logarithms that $\log_b(b^x) = x$ . $\newline$ Applying this property, we get $\log_6(6^{\frac{3}{5}}) = \frac{3}{5}$ .