Definition: Given observed data $X=x$, the likelihood function for $\theta$ is defined as
\[L(\theta \mid x) = f(x \mid \theta).\]Here, $f(x \mid \theta)$ can be a density function, point mass function, or a mixture of the two.
In practice, it is typically more sensible to work with the log-likelihood instead, which is simply the (natural) log of the likelihood:
\[\ell(\theta \mid x) = \log L(\theta \mid x).\]