Question 459621
{{{log(4,(y)) + log(y,(4))}}}{{{""=""}}}{{{5/2}}}
<pre>
Use the identity:

{{{log(A,(B))}}}{{{""=""}}}{{{1/log(B,(A))}}}, {{{0<A<>1}}}, {{{0<B<>1}}} 

on the second term:

{{{log(4,(y)) + 1/log(4,(y))}}}{{{""=""}}}{{{5/2}}}

Let {{{U=log(4,(y))}}}

{{{U+1/U=5/2}}}

Clear of fractions by multiplying through by 2U

{{{2U^2 + 2 = 5U}}}

Get 0 on the right:

{{{2U^2-5U+2=0}}} 

Factor the left side:

{{{(2U-1)(U-2)=0}}}

{{{2U-1=0}}}
{{{2U=1}}}
{{{U=1/2}}}

{{{U-2=0}}}
{{{U=2}}}

For {{{U=1/2}}}
Replace U by {{{U=log(4,(y))}}}

{{{log(4,y)=1/2}}}

Replace by the exponential equivalent:

{{{y=4^(1/2)=sqrt(4)=2}}}

For {{{U=2}}}
Replace U by {{{U=log(4,(y))}}}

{{{log(4,y)=2}}}

Replace by the exponential equivalent:

{{{y=4^2=16}}}

So there are two solutions, y = 2 and y = 16.

Edwin</pre>