Question 21503
Using logarithms:

{{{2^(1-x) = 3^x}}} Take the log of both sides.
{{{(1-x)log2 = xlog3}}} Expand the left side using the distributive property.
{{{log2 - xlog2 = xlog3}}} Add xlog2 to both sides.
{{{log2 = xlog3 + xlog2}}} Factor the x.
{{{log2 = x(log3 + log2)}}} Simplify the right side.(loga+logb=log(ab))
{{{log2 = xlog6}}} Divide both sides by log6
{{{x = (log2)/log6}}} Evaluate using your calculator or table of logarithms.
{{{x = 0.3868528}}}

Check:
{{{2^(1-0.3868528) = 2^0.6131472}}} = 1.5295923...
{{{3^0.3868528 = 1.5295923}}}