If `x gt 0 and log_(3) (x)+log_(3)(x^(1//3))+log_(3)(x^(1//9))+log_(3)(x^(1//27))+...=6`, then x=

Answer

To solve the equation

\[ \log_{3}(x) + \log_{3}(x^{1/3}) + \log_{3}(x^{1/9}) + \log_{3}(x^{1/27}) + \ldots = 6, \]

we can follow these steps:

### Step 1: Rewrite the logarithmic terms
Using the property of logarithms that states \(\log_{b}(a^c) = c \cdot \log_{b}(a)\), we can rewrite each term:

\[ \log_{3}(x) + \frac{1}{3} \log_{3}(x) + \frac{1}{9} \log_{3}(x) + \frac{1}{27} \log_{3}(x) + \ldots \]

This can be factored as:

\[ \log_{3}(x) \left(1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \ldots\right). \]

### Step 2: Identify the series
The series \(1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \ldots\) is a geometric series where the first term \(a = 1\) and the common ratio \(r = \frac{1}{3}\).

### Step 3: Sum the geometric series
The sum \(S\) of an infinite geometric series can be calculated using the formula:

\[ S = \frac{a}{1 - r} = \frac{1}{1 - \frac{1}{3}} = \frac{1}{\frac{2}{3}} = \frac{3}{2}. \]

### Step 4: Substitute back into the equation
Now substituting back into our equation, we have:

\[ \log_{3}(x) \cdot \frac{3}{2} = 6. \]

### Step 5: Solve for \(\log_{3}(x)\)
To isolate \(\log_{3}(x)\), we multiply both sides by \(\frac{2}{3}\):

\[ \log_{3}(x) = 6 \cdot \frac{2}{3} = 4. \]

### Step 6: Convert from logarithmic form to exponential form
Using the property of logarithms, we convert this to exponential form:

\[ x = 3^{4}. \]

### Step 7: Calculate the value of \(x\)
Calculating \(3^{4}\):

\[ x = 81. \]

Thus, the value of \(x\) is

\[ \boxed{81}. \]