Question

\(log_x2+log_x2^2+log_x2^3+………+ log_x2^n=\frac {n(n+1)}{2}\), then \(x=\)

• A. n
• B. 1
• C. 5
• D. 2

Answer 1

The Correct Option is D

To solve this problem, we need to determine the value of \( x \) in the given logarithmic expression.

1. Simplifying the Logarithmic Expression:
The given equation is: \[ \log_x 2 + \log_x 2^2 + \log_x 2^3 + \cdots + \log_x 2^n = \frac{n(n+1)}{2} \] Using the property of logarithms \( \log_x a^b = b \log_x a \), we can simplify each term: \[ \log_x 2 + 2\log_x 2 + 3\log_x 2 + \cdots + n\log_x 2 \] This simplifies to: \[ \log_x 2 (1 + 2 + 3 + \cdots + n) \] The sum of the first \( n \) natural numbers is given by: \[ 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2} \] Thus, the equation becomes: \[ \log_x 2 \times \frac{n(n+1)}{2} = \frac{n(n+1)}{2} \]

2. Solving for \( x \):
Since \( \log_x 2 \) appears on both sides of the equation, we can cancel out the \( \frac{n(n+1)}{2} \) term from both sides: \[ \log_x 2 = 1 \] The logarithmic equation \( \log_x 2 = 1 \) means that \( x^1 = 2 \), or \( x = 2 \).

Final Answer:
The correct answer is (D) 2.