Answer
$t = 4{\left( {\ln |x|} \right)^2}$
Work Step by Step
$$\eqalign{
& {\text{We have the equation}} \cr
& {e^{\sqrt t }} = {x^2} \cr
& {\text{To solve this equation for }}t,{\text{ first take natural logarithm }} \cr
& {\text{on both sides}} \cr
& \ln \left( {{e^{\sqrt t }}} \right) = \ln \left( {{x^2}} \right) \cr
& {\text{Apply the property }}\ln {a^n} = n\ln a \cr
& \sqrt t \ln \left( e \right) = 2\ln |x| \cr
& \sqrt t = 2\ln x \cr
& {\text{Now}}{\text{, square both sides of the equation}} \cr
& {\left( {\sqrt t } \right)^2} = {\left( {2\ln |x|} \right)^2} \cr
& {\text{Simplify}} \cr
& t = 4{\left( {\ln |x|} \right)^2} \cr} $$
