If `"log"_(8){"log"_(2) "log"_(3) (x^(2) -4x +85)} = (1)/(3)`, then x equals to

Options

A. 5
B. 4
C. 3
D. 2

Answer

To solve the equation \( \log_8(\log_2(\log_3(x^2 - 4x + 85))) = \frac{1}{3} \), we will follow these steps:

### Step 1: Rewrite the logarithmic equation
We start with the equation:
\[ \log_8(\log_2(\log_3(x^2 - 4x + 85))) = \frac{1}{3} \]
This means that:
\[ \log_2(\log_3(x^2 - 4x + 85)) = 8^{\frac{1}{3}} = 2 \]

### Step 2: Simplify the equation
Next, we simplify:
\[ \log_2(\log_3(x^2 - 4x + 85)) = 2 \]
This implies:
\[ \log_3(x^2 - 4x + 85) = 2^2 = 4 \]

### Step 3: Exponentiate to eliminate the logarithm
Now, we exponentiate:
\[ x^2 - 4x + 85 = 3^4 = 81 \]

### Step 4: Rearrange into a standard quadratic equation
Rearranging gives us:
\[ x^2 - 4x + 85 - 81 = 0 \]
Which simplifies to:
\[ x^2 - 4x + 4 = 0 \]

### Step 5: Factor the quadratic equation
This can be factored as:
\[ (x - 2)^2 = 0 \]

### Step 6: Solve for \( x \)
Thus, we find:
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]

### Conclusion
The value of \( x \) is \( 2 \).

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