Answer
$$ y = - 2{\log _{12}}3$$
Work Step by Step
$$\eqalign{
& {4^{ - y}} = {3^{y + 2}} \cr
& {\text{take the natural logarithm on both sides}} \cr
& \ln {4^{ - y}} = \ln {3^{y + 2}} \cr
& {\text{use the power property}} \cr
& - y\ln 4 = \left( {y + 2} \right)\ln 3 \cr
& - y\ln 4 = y\ln 3 + 2\ln 3 \cr
& {\text{solve for }}y \cr
& - y\ln 4 - y\ln 3 = 2\ln 3 \cr
& y\ln 4 + y\ln 3 = - 2\ln 3 \cr
& \left( {\ln 4 + \ln 3} \right)y = - 2\ln 3 \cr
& \left( {\ln 12} \right)y = - 2\ln 3 \cr
& y = \frac{{ - 2\ln 3}}{{\ln 12}} \cr
& y = - 2{\log _{12}}3 \cr} $$
