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

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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 )

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