For real x, which statements about y= e(-ex) are correct?

$y=x\pm in\pi$ for even integer n, i²=−1

GATE 2025 BM – Question 42

Engineering Mathematics · Complex Variables · 2 marks · Multiple select

For real x, which statements about $y=\log_e(-e^x)$ are correct?

y is real
y is complex
$y=x\pm in\pi$ for odd integer n, i²=−1
$y=x\pm in\pi$ for even integer n, i²=−1

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Correct answer: (B) y is complex; (C) $y=x\pm in\pi$ for odd integer n, i²=−1

Explanation

The argument −eˣ is a negative real number. Its complex arguments are (2k+1)π, so logarithm values are x+i(2k+1)π. They are nonreal and use odd multiples of π. Therefore B,C; a real logarithm would be undefined.

GATE 2025 BM – Question 42

Explanation

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