`log_(1/3) 81` is equal to

Options

A. `-27`
B. `-4`
C. `4`
D. `127`

Answer

To solve the logarithmic expression \( \log_{(1/3)} 81 \), we can follow these steps:

### Step 1: Rewrite the logarithm
We start by rewriting \( 81 \) and the base \( \frac{1}{3} \) in terms of powers of \( 3 \):
\[ 81 = 3^4 \quad \text{and} \quad \frac{1}{3} = 3^{-1} \]
Thus, we can rewrite the logarithm as:
\[ \log_{(1/3)} 81 = \log_{(3^{-1})} (3^4) \]

### Step 2: Apply the logarithmic property
Using the property of logarithms that states \( \log_{(b^m)} (a^n) = \frac{n}{m} \log_b a \), we can apply this here:
\[ \log_{(3^{-1})} (3^4) = \frac{4}{-1} \log_3 3 \]

### Step 3: Simplify using the logarithm of the same base
Since \( \log_3 3 = 1 \) (because the logarithm of a number to its own base is always 1), we can simplify further:
\[ \frac{4}{-1} \cdot 1 = -4 \]

### Final Answer
Thus, the value of \( \log_{(1/3)} 81 \) is:
\[ \boxed{-4} \]

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