The value of `""_(27)log_3 (81)` is:

Options

A. 27381
B. 531441
C. 65631
D. none of these

Answer

To solve the problem of finding the value of \(27^{\log_3(81)}\), we can follow these steps:

### Step 1: Simplify \(\log_3(81)\)
First, we need to find \(\log_3(81)\). We know that \(81\) can be expressed as a power of \(3\):
\[ 81 = 3^4 \]
Thus, we can rewrite the logarithm:
\[ \log_3(81) = \log_3(3^4) \]
Using the property of logarithms that states \(\log_b(a^c) = c \cdot \log_b(a)\), we have:
\[ \log_3(3^4) = 4 \cdot \log_3(3) \]
Since \(\log_3(3) = 1\), we find:
\[ \log_3(81) = 4 \cdot 1 = 4 \]

### Step 2: Substitute back into the expression
Now that we have \(\log_3(81) = 4\), we can substitute this back into our original expression:
\[ 27^{\log_3(81)} = 27^4 \]

### Step 3: Simplify \(27^4\)
Next, we need to calculate \(27^4\). We know that \(27\) can be expressed as \(3^3\):
\[ 27 = 3^3 \]
Therefore:
\[ 27^4 = (3^3)^4 \]
Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we can simplify this to:
\[ (3^3)^4 = 3^{3 \cdot 4} = 3^{12} \]

### Step 4: Calculate \(3^{12}\)
Now, we need to calculate \(3^{12}\). We can do this step by step:
\[ 3^2 = 9 \]
\[ 3^4 = 9^2 = 81 \]
\[ 3^8 = 81^2 = 6561 \]
Now, we multiply \(3^8\) and \(3^4\):
\[ 3^{12} = 3^8 \cdot 3^4 = 6561 \cdot 81 \]
Calculating \(6561 \cdot 81\):
\[ 6561 \cdot 81 = 531441 \]

### Final Answer
Thus, the value of \(27^{\log_3(81)}\) is:
\[ \boxed{531441} \]