Question 770321
Given 2^x=3^y=5^z=900^w 
Show that:
xyz=2w(xy+yz+xz)
<pre>
Suppose that 

2<sup>x</sup> = 3<sup>y</sup> = 5<sup>z</sup> = 900<sup>w</sup> = A

Then,

ln(2<sup>x</sup>) = ln(3<sup>y</sup>) = ln(5<sup>z</sup>) = ln(900<sup>w</sup>) = ln(A)

x·ln(2) = y·ln(3) = z·ln(5) = w·ln(900) = ln(A)

{{{x=ln(A)/ln(2)}}}, {{{y=ln(A)/ln(3)}}}, {{{z=ln(A)/ln(5)}}}, {{{w=ln(A)/ln(900)}}}

<font color="red">***</font> xyz = {{{ln(A)/ln(2)}}}{{{""*""}}}{{{ln(A)/ln(3)}}}{{{""*""}}}{{{ln(A)/ln(5)}}} = {{{(ln(A))^3/(ln(2)ln(3)ln(5))}}}<font color="red">***</font>       

xy = {{{(ln(A)/ln(2))}}}{{{""*""}}}{{{(ln(A)/ln(3))}}} = {{{(ln(A))^2/(ln(2)ln(3))}}}

yz = {{{(ln(A)/ln(3))}}}{{{""*""}}}{{{(ln(A)/ln(5))}}} = {{{(ln(A))^2/(ln(3)ln(5))}}}

xz = {{{(ln(A)/ln(2))}}}{{{""*""}}}{{{(ln(A)/ln(5))}}} = {{{(ln(A))^2/(ln(2)ln(5))}}}  

xy + yz + xz = {{{(ln(A))^2/(ln(2)ln(3))}}}{{{""+""}}}{{{(ln(A))^2/(ln(2)ln(3))}}}{{{""+""}}}{{{(ln(A))^2/(ln(2)ln(3))}}}{{{""=""}}}

          {{{(ln(A))^2( 1/(ln(2)ln(3)) + 1/(ln(3)ln(5))+ 1/(ln(2)ln(5)))}}}{{{""=""}}}{{{(ln(A))^2( ( ln(5) + ln(2)+ ln(3))/(ln(2)ln(3)ln(5)))}}}

{{{w}}}{{{""=""}}}{{{ln(A)/ln(900)}}}{{{""=""}}}{{{ln(A)/ln(2^2*3^2*5^2)}}}{{{""=""}}}{{{ln(A)/(ln(2^2)+ln(3^2)+ln(5^2))}}}{{{""=""}}}{{{ln(A)/(2ln(2)+2ln(3)+2ln(5))}}}{{{""=""}}}

          {{{ln(A)/(2(ln(2)+ln(3)+ln(5)))}}}

So 2w = {{{2ln(A)/(2(ln(2)+ln(3)+ln(5)))}}}{{{""=""}}}{{{cross(2)ln(A)/(cross(2)(ln(2)+ln(3)+ln(5)))}}}{{{""=""}}}{{{ln(A)/(ln(2)+ln(3)+ln(5))}}}   
    
2w(xy+yz+xz){{{""=""}}}{{{ln(A)/(ln(2)+ln(3)+ln(5))}}}{{{""*""}}}{{{(ln(A))^2( ( ln(5) + ln(2)+ ln(3))/(ln(2)ln(3)ln(5)))}}}{{{""=""}}}

          {{{ln(A)/(cross(ln(2)+ln(3)+ln(5)))}}}{{{""*""}}}{{{(ln(A))^2( (cross( ln(5) + ln(2)+ ln(3)))/(ln(2)ln(3)ln(5)))}}}{{{""=""}}}{{{(ln(A))^3/(ln(2)ln(3)ln(5))}}}

And that's what xyz was shown equal to above, marked <font color="red">***</font>      

Edwin</pre>