`log_(3)root(9)(3)=`

Options

A. `(1)/(9)`
B. `(3)/(2)`
C. `(1)/(2)`
D. 1

Explanation

To solve the problem \( \log_{3} \sqrt[9]{3} \), we can follow these steps:

### Step 1: Rewrite the expression
We start by rewriting the 9th root of 3 in exponential form:
\[ \sqrt[9]{3} = 3^{1/9} \]
Thus, we can express the logarithm as:
\[ \log_{3} \sqrt[9]{3} = \log_{3} (3^{1/9}) \]

### Step 2: Apply the power rule of logarithms
Using the logarithmic identity \( \log_{b} (a^{n}) = n \cdot \log_{b} (a) \), we can simplify:
\[ \log_{3} (3^{1/9}) = \frac{1}{9} \cdot \log_{3} (3) \]

### Step 3: Evaluate \( \log_{3} (3) \)
We know that \( \log_{b} (b) = 1 \) for any base \( b \). Therefore:
\[ \log_{3} (3) = 1 \]

### Step 4: Substitute back into the equation
Now substituting back, we have:
\[ \log_{3} (3^{1/9}) = \frac{1}{9} \cdot 1 = \frac{1}{9} \]

### Final Answer
Thus, the value of \( \log_{3} \sqrt[9]{3} \) is:
\[ \frac{1}{9} \]