The difference of roots of the equation `((log)_(27)x^3)^2=(log)_(27)x^6` is .......

Options

A. `(2)/(3)`
B. 1
C. 9
D. 8

Answer

To solve the equation \(((\log_{27} x^3)^2 = \log_{27} x^6\), we will follow these steps:

### Step 1: Rewrite the equation using logarithmic properties
We know that \(\log_{27} x^3 = 3 \log_{27} x\) and \(\log_{27} x^6 = 6 \log_{27} x\). Let \(y = \log_{27} x\). Therefore, we can rewrite the equation as:
\[ (3y)^2 = 6y \]

### Step 2: Simplify the equation
Expanding the left side gives us:
\[ 9y^2 = 6y \]
Now, we can rearrange this to form a standard quadratic equation:
\[ 9y^2 - 6y = 0 \]

### Step 3: Factor the quadratic equation
Factoring out \(y\) from the equation:
\[ y(9y - 6) = 0 \]
This gives us two possible solutions:
1. \(y = 0\)
2. \(9y - 6 = 0 \Rightarrow y = \frac{6}{9} = \frac{2}{3}\)

### Step 4: Solve for \(x\) using the values of \(y\)
Now we will convert back to \(x\) using the values of \(y\):

1. For \(y = 0\):
\[ \log_{27} x = 0 \Rightarrow x = 27^0 = 1 \]

2. For \(y = \frac{2}{3}\):
\[ \log_{27} x = \frac{2}{3} \Rightarrow x = 27^{\frac{2}{3}} = (3^3)^{\frac{2}{3}} = 3^2 = 9 \]

### Step 5: Find the difference between the roots
The two roots we found are \(x = 1\) and \(x = 9\). The difference between these roots is:
\[ 9 - 1 = 8 \]

### Final Answer
The difference of the roots of the equation is \(8\).

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