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

Explanation

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

### Step 1: Rewrite the logarithms
We can use the change of base formula and properties of logarithms to simplify the equation.

\[ \log_{27} x^3 = \frac{3}{3} \log_3 x = \log_3 x \]
\[ \log_{27} x^6 = \frac{6}{3} \log_3 x = 2 \log_3 x \]

### Step 2: Substitute the logarithmic expressions into the equation
Now, substitute these expressions back into the original equation:

\[ (\log_3 x)^2 = 2 \log_3 x \]

### Step 3: Rearrange the equation
Rearranging gives us:

\[ (\log_3 x)^2 - 2 \log_3 x = 0 \]

### Step 4: Factor the equation
We can factor the left-hand side:

\[ \log_3 x (\log_3 x - 2) = 0 \]

### Step 5: Solve for \(\log_3 x\)
Setting each factor to zero gives us two equations:

1. \(\log_3 x = 0\)
2. \(\log_3 x - 2 = 0\)

### Step 6: Solve each equation
For the first equation:

\[ \log_3 x = 0 \implies x = 3^0 = 1 \]

For the second equation:

\[ \log_3 x = 2 \implies x = 3^2 = 9 \]

### Step 7: Find the difference of the roots
The roots we found are \(x = 1\) and \(x = 9\). The difference of the roots is:

\[ 9 - 1 = 8 \]

### Final Answer
The difference of the roots of the equation is **8**.

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