Answer
$\displaystyle \{\frac{\log 5-\log 2}{\log 2+2\log 5}\}$ or $\{0.234\}$
Work Step by Step
... Apply $\log$(...) to both sides
$\log 2^{x+1}=\log 5^{1-2x}$ $\quad $...Apply$: \quad \log_{a}M^{r}=r\log_{a}M\quad $
$(x+1)\log 2=(1-2x)\log 5\quad $... isolate x
$x\log 2+\log 2=\log 5-2x\log 5$
$x\log 2+2x\log 5==\log 5-\log 2$
$x(\log 2+2\log 5)=\log 5-\log 2$
$x=\displaystyle \frac{\log 5-\log 2}{\log 2+2\log 5}\approx 0.234$
Solution set: $\displaystyle \{\frac{\log 5-\log 2}{\log 2+2\log 5}\}$ or $\{0.234\}$