` If log_2 (log_8x)=log_8(log_2x),` find the value of `(log_2x)^2.`

Options

A. 1
B. 27
C. 3
D. Not defined

Explanation

To solve the equation \( \log_2 (\log_8 x) = \log_8 (\log_2 x) \), we will follow these steps:

### Step 1: Rewrite the logarithms in terms of a common base
We know that \( \log_8 x \) can be rewritten using the change of base formula:
\[ \log_8 x = \frac{\log_2 x}{\log_2 8} = \frac{\log_2 x}{3} \]
because \( 8 = 2^3 \).

### Step 2: Substitute into the equation
Substituting this back into our original equation gives:
\[ \log_2 \left(\frac{\log_2 x}{3}\right) = \log_8 (\log_2 x) \]

### Step 3: Rewrite the right-hand side
Now we rewrite the right-hand side:
\[ \log_8 (\log_2 x) = \frac{\log_2 (\log_2 x)}{3} \]
So we have:
\[ \log_2 \left(\frac{\log_2 x}{3}\right) = \frac{\log_2 (\log_2 x)}{3} \]

### Step 4: Simplify the left-hand side
Using the property of logarithms, we can simplify the left-hand side:
\[ \log_2 \left(\frac{\log_2 x}{3}\right) = \log_2 (\log_2 x) - \log_2 3 \]
Thus, our equation becomes:
\[ \log_2 (\log_2 x) - \log_2 3 = \frac{\log_2 (\log_2 x)}{3} \]

### Step 5: Clear the fraction
To eliminate the fraction, multiply the entire equation by 3:
\[ 3 \log_2 (\log_2 x) - 3 \log_2 3 = \log_2 (\log_2 x) \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ 3 \log_2 (\log_2 x) - \log_2 (\log_2 x) = 3 \log_2 3 \]
This simplifies to:
\[ 2 \log_2 (\log_2 x) = 3 \log_2 3 \]

### Step 7: Solve for \( \log_2 (\log_2 x) \)
Dividing both sides by 2:
\[ \log_2 (\log_2 x) = \frac{3}{2} \log_2 3 \]

### Step 8: Exponentiate to remove the logarithm
Exponentiating both sides gives:
\[ \log_2 x = 3^{\frac{3}{2}} = 3\sqrt{3} \]

### Step 9: Find \( (\log_2 x)^2 \)
Now, we need to find \( (\log_2 x)^2 \):
\[ (\log_2 x)^2 = (3\sqrt{3})^2 = 27 \]

Thus, the value of \( (\log_2 x)^2 \) is \( \boxed{27} \).