The likelihood function is the probability of the parameters $\theta$ given the data $(x_1,\dots,x_n)$:

$$ L(\theta)=\prod^n_{i=1}p(x_i\mid\theta) $$

Log-likelihood function

The log-likelihood function is defined as:

$$ \ell(\theta)=\sum^n_{i=1}\log p(x_i\mid\theta) $$

The log-likelihood is often used in practice because of better numerical stability.