  We present black hole solutions in $2+1-$dimensional Einstein's theory of
gravity coupled with Born-Infeld nonlinear electrodynamic and a massless
self-interacting scalar field. The model has five free parameters: mass $M$,
cosmological constant $\ell $, electric $q$ and scalar $r_{0}$ charges and
Born-Infeld parameter $\beta $. To attain exact solution for such a highly
non-linear system we adjust, i.e. finely tune, the parameters of the theory
with the integration constants. In the limit $\beta \rightarrow 0$ we recover
the results of Einstein-Maxwell-Scalar theory, obtained before. The self
interacting potential admits finite minima apt for the vacuum contribution.
Hawking temperature of the model is investigated versus properly tuned
parameters. By employing this tuned-solution as basis, we obtain also a dynamic
solution which in the proper limit admits the known solution in Einstein
gravity coupled with self-interacting scalar field. Finally we establish the
equations of a general scalar-tensor field coupled to nonlinear electrodynamics
field\ in $2+1-$dimensions without searching for exact solutions.
