Question

Taking Log both siden,
( Rightarrow frac{log _{x}(2009) x}{2010}=log (2009)^{log _{x}(2010)} )
( Rightarrow log _{x} x+log _{x} 2009-log _{x} 2010 )
( = )
( =log _{x}(2010), log 2009 )
( Rightarrow 1-operatorname{tag} x^{20} 10+log _{x} 2009left(1-log _{x} 2010right)=0 )
( Rightarrow log _{x} 200 y=-frac{left(1-log _{x}(2010)right)}{left(1-log _{x}(2010)right)} )
( Rightarrow log _{x} 2009=-1 )
( Rightarrow quad 2009=x^{-1} )
( Rightarrow quad 2009=frac{1}{x} )
( Rightarrow quad x=frac{1}{2009}=frac{m}{n} )
( Rightarrow m-n>=(1-2009) )
( =-2008 )

# Sectior F.161 162. If the product of all solutions of the equation - (2009) 2010 form as then the value of ( m n ) is can be expressed in the lowest m

Solution