Answer
$x=-\dfrac{\log3+\log5}{2\log5-4\log3}\approx2.303599$
Work Step by Step
$5^{2x+1}=3^{4x-1}$
Apply $\log$ to both sides of the equation:
$\log5^{2x+1}=\log3^{4x-1}$
Take the exponents down to multiply in front of their respective $\log$:
$(2x+1)\log5=(4x-1)\log3$
Evaluate the products on both sides:
$2x\log5+\log5=4x\log3-\log3$
Take $4x\log3$ to the left side and $\log5$ to the right side:
$2x\log5-4x\log3=-\log3-\log5$
Take out common factor $x$ from the left side:
$x(2\log5-4\log3)=-\log3-\log5$
Solve for $x$:
$x=-\dfrac{\log3+\log5}{2\log5-4\log3}\approx2.303599$
