New-Keynesian Model: Log-linearizing the firm's FOC
$$(\sum_{k=0}^\infty \theta^k E_k(Q_{t,t+k} Y_{t+k|t} (P_t^*/P_{t-1} - \alpha MC_{t,t+k} \beta_{t-1,t+k}))) = 0 $$

Log linearize around the zero inflation steady state.

$$p_t^* - p_{t-1} = (1 - \beta\theta) \sum_{k=0}^\infty (\beta\theta)^k E_t(\widehat{mc}_{t+k|t} + p_{t+k} - p_{t-1}) $$

Where $$\widehat{mc}_{t+k|t} \equiv {mc}_{t+k|t} - mc$$
In other words,
$$p_t^* = \gamma + (1 - \beta\theta) \sum_{k=0}^\infty (\beta\theta)^k E_t(\widehat{mc}_{t+k|t} + p_{t+k}) $$
where $\gamma \equiv log \frac{\epsilon}{\epsilon + 1} $

If I remember correctly, this model uses Calvo pricing, so inflation is just from wage stickiness. So if you set $\theta = 0$ (no price stickiness)
$$p_t^* = \gamma + mc_t + p_t$$

See what you can work with from there.

Edit: If you're wondering how this is done, then "An Old Man in the Sea"'s resource in the comments is helpful, in sections 3.2 and 3.3. It will show how to do the First-Order Taylor Expansion to derive the result.