How do I get the Likelihood of this Bayesian parameter $\theta$ in the following conditional equation?

Question

The intenisty (hazard) function is:
$\lambda$(t|$\theta$)

And the function of t is:
$f(t| \theta)$ = $\dfrac{\lambda(t|\theta)}{1-S(t)}$
Where $1-S(t)$ is $F(t) = P(0 < t < T]$

The log likelihood is: (what I want to get)
\begin{equation} \begin{split} \textrm{log}\mathcal{L} (\theta) & = \sum_{i=1}^{N} \textrm{log}\lambda(t_i - \theta) - \int_{0}^{T} \lambda(t|\theta)dt \end{split} \end{equation}

This is taken from the Seismic data Analysis for Taiwan by Fan and Kuo, they show those equations but they don't explain how they got there. I have tried Bayes rule to get $\lambda(\theta|t)$ so I can get the Likelihood, but what I get is:

From Bayes:
$\lambda(\theta|t) = \dfrac{\lambda(t|\theta) f(\theta)}{f(t)} $

Replacing $f(t)$ by the above equation (and doing a bit of algebra):

$\lambda(\theta|t) = \dfrac{\lambda(t|\theta)f(\theta)}{\lambda(\theta)} - \dfrac{\lambda(t|\theta) f(\theta) S(t)}{\lambda(\theta)}$

Am I getting somewhere or am I even on the right track? Any advice anywone? Thanks for ur time!