Answer
$x\approx20.67$
Work Step by Step
Use the definition of the log function to solve for $x$. Rewrite the problem so that the base has the exponent of the opposite side and is equal to the log's number and/or variable.
$\log_4(3x+2)=3$
$4^3=3x+2$
$4^3-2=3x$
$x=\frac{62}{3}$
$x\approx20.67$