The value of `2logx+2logx^(2)+2logx^(3)+....n` terms `(xgt0)` is …..

Answer

To solve the problem, we need to find the value of the expression \(2 \log x + 2 \log x^2 + 2 \log x^3 + \ldots\) for \(n\) terms, where \(x > 0\).

### Step-by-Step Solution:

1. **Rewrite the Expression**:
The given expression can be rewritten using the properties of logarithms:
\[ 2 \log x + 2 \log x^2 + 2 \log x^3 + \ldots + 2 \log x^n \]
This can be factored as:
\[ 2 (\log x + \log x^2 + \log x^3 + \ldots + \log x^n) \]

2. **Apply Logarithm Properties**:
Using the property of logarithms that states \(\log a^b = b \log a\), we can rewrite each term:
\[ \log x^k = k \log x \]
Thus, the expression becomes:
\[ 2 (\log x + 2 \log x + 3 \log x + \ldots + n \log x) \]

3. **Factor Out \(\log x\)**:
We can factor out \(\log x\) from the sum:
\[ 2 \log x (1 + 2 + 3 + \ldots + n) \]

4. **Sum of the First \(n\) Natural Numbers**:
The sum of the first \(n\) natural numbers is given by the formula:
\[ S_n = \frac{n(n + 1)}{2} \]
Therefore, we can substitute this into our expression:
\[ 2 \log x \cdot \frac{n(n + 1)}{2} \]

5. **Simplify the Expression**:
The \(2\) in the numerator and denominator cancels out:
\[ n(n + 1) \log x \]

### Final Answer:
Thus, the value of the expression \(2 \log x + 2 \log x^2 + 2 \log x^3 + \ldots\) for \(n\) terms is:
\[ \boxed{n(n + 1) \log x} \]