What is the value of log, base, 6, cube root of, 6, end cube rootlog 6​ 3 6​ ?

To solve for the value of $\log_6 \left( \sqrt[3]{6} \right)$, we can break it down using logarithmic properties.

1. ### Break Down the Problem

We need to compute $\log_6 \left( \sqrt[3]{6} \right)$. The expression $\sqrt[3]{6}$ can be rewritten as $6^{1/3}$.

2. ### Relevant Concepts

Using the property of logarithms: $\log_b (a^c) = c \cdot \log_b(a)$ we can express our logarithm as follows: $\log_6 \left( \sqrt[3]{6} \right) = \log_6 \left( 6^{1/3} \right)$

3. ### Analysis and Detail

Now applying the logarithmic property: $\log_6 \left( 6^{1/3} \right) = \frac{1}{3} \cdot \log_6(6)$ Since $\log_6(6) = 1$, we can simplify further: $\frac{1}{3} \cdot \log_6(6) = \frac{1}{3} \cdot 1 = \frac{1}{3}$

4. ### Verify and Summarize

The calculations are verified, and the final value computed is consistent.

Final Answer

Thus, the value of $\log_6 \left( \sqrt[3]{6} \right)$ is: $\frac{1}{3}$