Question

\(\log x + \log x^3 + \log x^5 + \dots + \log x^{2n-1}\) is equal to

• A. \(2n \log x\)
• B. \((2n-1)\log x\)
• C. \(n^2 \log x\)
• D. \((n^2+1)\log x\)

Hint:

Sum of first n odd numbers = \(n^2\).

Answer 1

The Correct Option is C

Concept: \[ \log x^k = k \log x \]

Step 1: Rewrite.
\[ = (1+3+5+\dots+(2n-1))\log x \]

Step 2: Sum of odd numbers.
\[ 1+3+5+\dots+(2n-1) = n^2 \] \[ \Rightarrow n^2 \log x \]