The Problem: Assume $\mu$ is known. Show this is an exponential family with parameter $\sigma$.
The Attempt: If I can show that the function $f(x| \sigma)$ can be written as $a(\sigma)t(x) +b(\sigma) +r(x) $. If I take the natural log of the function, this looks like
$ln(f(x|\sigma)) = ln(\frac{1}{\sqrt{2\pi} \sigma} exp[\frac{-(x-\mu)^2}{2\sigma^2}]) = -ln(\sqrt{2\pi} \sigma) - \frac{(x-\mu)^2}{2\sigma^2}$.
This is all I have so far. I tried completing the square for the second term but I did not get as far as I expected.
Is there something I am missing? Thank you for your help!