Answer
$x=\dfrac{\log7}{3\log2}+\dfrac{5}{3}\approx2.60$
Work Step by Step
$2^{3x-5}=7$
Apply $\log$ to both sides of the equation:
$\log2^{3x-5}=\log7$
Take $3x-5$ to multiply in front of its $\log$:
$(3x-5)\log2=\log7$
Solve for $x$:
$3x-5=\dfrac{\log7}{\log2}$
$3x=\dfrac{\log7}{\log2}+5$
$x=\dfrac{\log7}{3\log2}+\dfrac{5}{3}\approx2.60$