Answer
$\displaystyle \{\frac{\log 3}{\log 36}\}$ or $\{0.307\}$
Work Step by Step
... Apply $\log$(...) to both sides
$\log 3^{1-2x}=\log 4^{x}$ $\quad $...Apply$: \quad \log_{a}M^{r}=r\log_{a}M\quad $
$(1-2x)\log 3=x\log 4 \quad $... isolate x
$\log 3-2x\log 3=x\log 4$
$\log 3=x\log 4+2x\log 3$
$\log 3=x(\log 4+2\log 3) \quad $...Apply$: \quad \log_{a}M^{r}=r\log_{a}M\quad $
$\log 3=x(\log 4+\log 9) \quad $...Apply$: \quad \log_{a}(MN)=\log_{a}M+\log_{a}N$
$\log 3=x(\log 36)$
$x=\displaystyle \frac{\log 3}{\log 36}\approx 0.307$
Solution set: $\displaystyle \{\frac{\log 3}{\log 36}\}$ or $\{0.307\}$
