Question 1152654
{{{2log(x)=x/25}}}

{{{2*25log(x)=x}}}

{{{log(x)=x/50}}}....................if {{{f (x )=g (x )}}}, then {{{a^(f (x ))=a^(g(x))}}}

{{{10^(log(10,x))=10^(x/50)}}}.......simplify : {{{10^(log(10,x))= x}}}

{{{x=10^(x/50)}}}...............use  Lambert form:  {{{xe^(- ln (10 )x/50))=1}}}


rewrite the equation with :{{{ u=-(ln (10 )x)/50}}}  and {{{x= -50u/ln (10)}}}

 
{{{(- 50u/ln (10))e^u=1}}}=>  in Lambert form is {{{e^u*u=-(ln(10)/50)}}}


solve  {{{e^u*u=-ln(10)/50)}}}


{{{u=-2ln(10)}}}, {{{u= W[0](-ln(10)/50)}}}


substitute back :{{{u=-(ln(10)x)/50)}}}, solve for {{{x}}}

{{{-(ln (10 )x)/50=-2*ln(10)}}}

{{{-(ln (10 ))x=-2*50*ln(10)}}}

{{{-ln (10 )x=-100*ln(10)}}}

{{{x=(-100*ln(10))/(-ln (10))}}}

{{{x=100}}}


solve  {{{-(ln (10 ))x/50= W[0](-ln(10)/50)}}}

{{{-(ln (10 ))x=50W[0]((-ln(10))/50))}}}


{{{x=50W[0]((-ln(10))/50)/(-(ln (10 ))))}}}


{{{x=-(50W[0](-ln(10)/50))/ln (10)}}}


{{{x=-(50W[0](-1/50))}}}


{{{x=50W[0]/50}}}


The solutions are:


{{{x=100}}}, {{{x=50W[0]/50}}}