What is the value of log, base, 9, cube root of, 81, end cube rootlog 9​ 3 81​ ?

1. Break Down the Problem

We need to calculate $\log_9(\sqrt[3]{81})$.

2. Relevant Concepts

To solve this, we can use properties of logarithms and exponents:

• The expression $\sqrt[3]{81}$ can be simplified.
• We also utilize the change of base formula if necessary.

3. Analysis and Detail

1. First, simplify $\sqrt[3]{81}$: $81 = 3^4 \implies \sqrt[3]{81} = 81^{1/3} = (3^4)^{1/3} = 3^{4/3}$

2. Now we find $\log_9(3^{4/3})$: Using the logarithm power rule, we have: $\log_9(3^{4/3}) = \frac{4}{3} \log_9(3)$

3. To compute $\log_9(3)$, we can express it using the change of base formula: $\log_9(3) = \frac{\log_3}{\log_9} = \frac{1}{2}$ because $9 = 3^2$.

4. Substitute back into the logarithm expression: $\log_9(3^{4/3}) = \frac{4}{3} \cdot \frac{1}{2} = \frac{2}{3}$

4. Verify and Summarize

The calculations have been traced back through each step to ensure accuracy. Therefore,

Final Answer

The value of $\log_9(\sqrt[3]{81})$ is $\frac{2}{3}$.