I’m having trouble log-linearising this equation - 1/C_t = \alpha (M_t^{-\sigma}) + \beta (1/(C_{t+1}V_{t+1}))

I conducted the first order Taylor approximation and then tried to expand the fractions, but I’m still not able to really get a satisfactory answer, especially because I can’t figure out how to remove the sigma as an exponent.

the most I’m able to get to is -\sigma \alpha M^{-\sigma} \hat m_t + (\beta/C_{t+1}V_{t+1}) \hat c_t

Yes, I tried some alternative steps I found in other resources, and no matter what, I can’t get the -SIGMA exponent to disappear. That can’t be used, can it?
I did see some papers which listed ‘linearised’ equations yet these equations had exponents as well. Im not sure if I should just proceed as is

You are confusing something here. Only the objects with a time index are variables. The rest are parameters. Thus, the equation is actually linear despite the exponents.