Question 145524
Let's evaluate f(g(x))


{{{f(x)=log(2,(x))}}} Start with the first function



{{{f(g(x))=log(2,(2^x))}}} Plug in {{{g(x)=2^x}}}



{{{f(g(x))=x*log(2,(2))}}} Rewrite the right side using the identity  {{{log(b,(x^y))=y*log(b,(x))}}}



{{{f(g(x))=x*1}}} Take the log base 2 of 2 to get 1. Note: {{{log(x,(x))=1}}}



{{{f(g(x))=x}}} Multiply




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Let's evaluate g(f(x))


{{{g(x)=2^x}}} Start with the second function



{{{g(f(x))=2^(log(2,(x)))}}} Plug in {{{f(x)=log(2,(x))}}}




Let {{{y=log(2,(x))}}}. So this means that {{{2^y=x}}} by use of the property {{{log(b,(x))=y}}} ====> {{{b^y=x}}}



{{{g(f(x))=2^y}}} Replace {{{log(2,(x))}}} with "y"



{{{g(f(x))=x}}} Now replace {{{2^y}}} with "x"





So <b>both</b> f(g(x)) and g(f(x)) equal x.