= Solution
Assume independent individuals and <independent censoring>, with the censoring law not involving the event-rate parameters. If $T_j$ is the <survival time> and $C_j$ its censoring time, we observe $x_j=\min(T_j,C_j)$ and $v_j=\mathbf1_{\{T_j\leq C_j\}}$. The <exponential distribution> gives
$$
f_j(x)=\theta_j e^{-\theta_jx},\qquad S_j(x)=e^{-\theta_jx}.
$$
An observed event contributes $f_j(x_j)$, while a <right-censored> observation contributes $S_j(x_j)$, because it only says the event has not occurred by $x_j$. Multiplying these <survival likelihood> contributions and omitting censoring factors independent of the parameters gives
$$
L(\theta_1,\ldots,\theta_n)\propto\prod_{j=1}^n\theta_j^{v_j}e^{-\theta_jx_j},
\qquad
\boxed{\ell(\theta)=\sum_{j=1}^n\{v_j\log\theta_j-\theta_jx_j\}+\text{constant}.}
$$
Without <independent censoring> the censoring factors cannot generally be discarded in this way.
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