Answer
Solution set = $\displaystyle \{\frac{2\ln 3-\ln 7}{2\ln 7-\ln 3}\}.$
$ x\approx 0.09$
Work Step by Step
$ 7^{2x+1}=3^{x+2}\qquad $... apply ln( ) to both sides
$\ln 7^{2x+1}=\ln 3^{x+2}\qquad $... apply the power rule for logarithms
$(2x+1)\ln 7=(x+2)\ln 3 \qquad $... distribute
$2x\ln 7+\ln 7=x\ln 3 +2\ln 3 \qquad $...isolate $x$
$ 2x\ln 7-x\ln 3=2\ln 3-\ln 7\qquad $... factor x out
$ x(2\ln 7-\ln 3)=2\ln 3-\ln 7\qquad $... divide with $(2\ln 7-\ln 3)$
$ x=\displaystyle \frac{2\ln 3-\ln 7}{2\ln 7-\ln 3}\qquad $... round to 2 decimal places
$ x\approx 0.09$
