Question 80792
{{{(2/5)^x = 7^(1-x)}}}


{{{log((2/5)^x) = log(7^(1-x))}}} Take the log base 10 of both sides(in this case, it doesn't matter which base we use) 


{{{xlog(2/5) = (1-x)log(7)}}} Use the identity {{{log(x^y)=ylog(x)}}}


{{{xlog(2/5) = log(7)-xlog(7)}}} Distribute log(7) to (1-x)


{{{xlog(2/5)+xlog(7) = log(7)}}} Get all x terms to one side


{{{x(log(2/5)+log(7)) = log(7)}}} Factor out an x


{{{x = log(7)/(log(2/5)+log(7))}}} Divide both sides by {{{log(2/5)+log(7)}}} to isolate x


So you are correct the answer is 


{{{x = log(7)/(log(2/5)+log(7))}}} or {{{x=1.88993148010236}}}



Check:

Plug in {{{x=1.88993148010236}}} to check

{{{(2/5)^1.88993148010236 = 7^(1-1.88993148010236)}}}


{{{0.1769785572734 = 7^(-0.88993148010236)}}}

{{{0.1769785572734 = 0.1769785572734}}} works