Answer
$x=\dfrac{\log3}{\log4}-7\approx-6.2075$
Work Step by Step
$4^{x+7}=3$
Apply $\log$ to both sides of the equation:
$\log4^{x+7}=\log3$
Take $x+7$ down to multiply in front of its respective $\log$:
$(x+7)\log4=\log3$
Solve for $x$:
$x+7=\dfrac{\log3}{\log4}$
$x=\dfrac{\log3}{\log4}-7\approx-6.2075$