Question 132314
{{{ 5^x=2^(x-6) }}} Start with the given equation



{{{ log(10,(5^x))=log(10,(2^(x-6))) }}} Take the log of both sides



{{{ x*log(10,(5))=(x-6)*log(10,(2)) }}} Use the identity {{{log(b,(x^y))=y*log(b,(x))}}} to rewrite both sides



{{{ x*log(10,(5))=x*log(10,(2))-6*log(10,(2)) }}} Distribute {{{log(10,(2))}}} (note: you distributed the terms incorrectly)



{{{ x*log(10,(5))-x*log(10,(2))=-6*log(10,(2)) }}} Subtract  {{{x*log(10,(2))}}} from both sides



{{{ x(log(10,(5))-log(10,(2)))=-6*log(10,(2)) }}} Factor out the GCF x



{{{ x(log(10,(5/2)))=-6*log(10,(2)) }}} Combine the logs on the left side using the identity {{{log(b,(A))-log(b,(B))=log(b,(A/B))}}}



{{{ x=-6*log(10,(2))/log(10,(5/2)) }}} Divide both sides by {{{log(10,(5/2))}}} to isolate x


Now if you want an approximate answer, then simply use your calculator to get 




{{{x=-4.53882}}}




So our answer is approximately {{{x=-4.53882}}}