before [@TOLNLED]. Recently, one of us considered BI type Lagrangians to obtain the black hole solutions [@HendiJHEP; @HendiAnn]. The Lagrangians of exponential and logarithmic forms of BI type theories may be defined as $$\mathcal{L}(\mathcal{F})=\left\{
\begin{array}{ll}
\beta ^{2}\left( \exp (-\frac{\mathcal{F}}{\beta ^{2}})-1\right) , & \;~%
\text{{ENED}} \\
-8\beta ^{2}\ln \left( 1+\frac{\mathcal{F}}{8\beta ^{2}}\right) , & \;\text{~%
{LNED}}%
\end{array}%
\right. ,  \label{LF}$$where $\beta $ is called the nonlinearity parameter, the Maxwell invariant is $\mathcal{F}=F_{\mu \nu }F^{\mu \nu }$ in which $F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }$ is the Faraday tensor and $A_{\mu } $ is the gauge potential. Although BI type models was introduced with various motivations, the important motivation of considering the BI type NED theories comes from the fact that these theories may be originated if one regards the loop corrections [@Fradkin85]. Taking into account the coupling of BI type theories with Einstein and Gauss-Bonnet gravity, it was shown that although there are some analogies between the BI type theories, there exist some differences between them as well [HendiJHEP,HendiAnn,HendiNon,HendiMahmudi]{}. Now, we would like to obtain new topological black hole solutions of the mentioned models of BI type theories coupled with third order Lovelock gravity and investigate their geometric and 