MLE of a Poisson distribution derivations


If we observe data ${{x_t}}^n_t$ from the model, $X_t sim Pois(tmu)$ independently.

Then how would I derive the maximum likelihood estimate of parameter $mu$?

So far, I have done this:

Let $$X_t sim Pois(lambda_t)$$

Then $$f(x_t, tmu) = frac{tmu^{x_t}exp(-tmu)}{x_t!}$$

Thus I determined the Likelihood function to be

$$ L(mu) = frac{tu sum_{t=1}^{n} x_t exp(-tmu)}{prod_{t=1}^{n}x_t}$$

So my next step would be finding the log liklihood. However I am confused as to what the notation for the log likelihood function would be. Would I be taking its as $l(mu)$ since $mu$ is the unknown parameter?

Thus I got $$ l(mu)= log (tmu)sum_{t=1}^{n} x_t – ntmu +c$$

How would I derive the log likelihood function to give the MLE? Also would that mean that the expectation of the MLE would just be $mu$? Thank you in advance