The equation `(log_(8)((8)/(x^(2))))/((log_(8)x)^(2))=3` has

Options

A. no integral solution
B. one natural solution
C. two real solutions
D. one irrational solution

Explanation

To solve the equation

\[ \frac{\log_8\left(\frac{8}{x^2}\right)}{(\log_8 x)^2} = 3, \]

we will follow these steps:

### Step 1: Rewrite the logarithm
Using the property of logarithms that states \(\log_a \left(\frac{A}{B}\right) = \log_a A - \log_a B\), we can rewrite the left-hand side:

\[ \log_8\left(\frac{8}{x^2}\right) = \log_8 8 - \log_8 x^2. \]

Since \(\log_8 8 = 1\) and \(\log_8 x^2 = 2 \log_8 x\), we have:

\[ \log_8\left(\frac{8}{x^2}\right) = 1 - 2 \log_8 x. \]

### Step 2: Substitute the expression into the equation
Substituting back into the equation gives:

\[ \frac{1 - 2 \log_8 x}{(\log_8 x)^2} = 3. \]

### Step 3: Clear the fraction
Multiply both sides by \((\log_8 x)^2\):

\[ 1 - 2 \log_8 x = 3 (\log_8 x)^2. \]

### Step 4: Rearrange the equation
Rearranging the equation gives:

\[ 3 (\log_8 x)^2 + 2 \log_8 x - 1 = 0. \]

### Step 5: Let \( t = \log_8 x \)
Substituting \( t \) for \(\log_8 x\), we have:

\[ 3t^2 + 2t - 1 = 0. \]

### Step 6: Solve the quadratic equation
Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

Here, \( a = 3, b = 2, c = -1 \):

\[ t = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 3 \cdot (-1)}}{2 \cdot 3} = \frac{-2 \pm \sqrt{4 + 12}}{6} = \frac{-2 \pm \sqrt{16}}{6} = \frac{-2 \pm 4}{6}. \]

Calculating the two possible values of \( t \):

1. \( t = \frac{2}{6} = \frac{1}{3} \)
2. \( t = \frac{-6}{6} = -1 \)

### Step 7: Convert back to \( x \)
Now substituting back for \( x \):

1. For \( t = \frac{1}{3} \):
\[ \log_8 x = \frac{1}{3} \implies x = 8^{\frac{1}{3}} = 2. \]

2. For \( t = -1 \):
\[ \log_8 x = -1 \implies x = 8^{-1} = \frac{1}{8}. \]

### Step 8: Conclusion
Thus, the solutions for \( x \) are \( x = 2 \) and \( x = \frac{1}{8} \).

### Final Answer
The equation has **two real solutions**: \( x = 2 \) and \( x = \frac{1}{8} \).

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