Question 948326
{{{log (y,x) = 3}}}...change the base to base {{{10}}}

{{{log (x)/log (y) = 3}}}

{{{log (x) = 3log (y)}}}

{{{log (x) = log (y^3)}}}...if log same, then we have

{{{x = y^3}}} ...........eq.1

{{{log (y,(4x)) = 5}}}

{{{log ((4x))/log (y) = 5}}}

{{{log ((4x)) = 5log (y)}}}

{{{log ((4x)) = log (y^5)}}}=>

{{{4x = y^5}}}

{{{x = y^5/4}}}..............eq.2

system is:

{{{x = y^3}}} ...........eq.1
{{{x = y^5/4}}}..............eq.2

since left sides same, right sides must be equal too

{{{y^3=y^5/4}}}...divide by {{{y^3}}}

{{{4y^3/y^3=y^5/y^3}}}

{{{4=y^2}}}

{{{y=sqrt(4)}}}

solutions:

=>{{{y=2}}} or {{{y=-2}}}


now find {{{x}}}

{{{x = y^3}}} ...........if {{{y=2}}}

{{{x = 2^3}}}

{{{x = 8}}}

{{{x = y^3}}} ...........if {{{y=-2}}}

{{{x = (-2)^3}}}

{{{x = -8}}}


intersection points: 
({{{0}}},{{{0}}})
({{{8}}},{{{2}}})
({{{-8}}},{{{-2}}})

check if all satisfy log

since we need {{{log (y,x) = 3}}}=> {{{log (0,0) <> 3}}}-this is not solution we can use


and {{{log (y,x) = 3}}}=> {{{highlight(log (2,8) = 3)}}}-this {{{is}}} solution we can use 

and {{{log (y,x) = 3}}}=> {{{log (2,-8) <> 3}}}-this is not solution we can use either



{{{drawing( 600, 600, -5, 10, -10, 10,
circle(8,2,.114),locate(8,2,p(8,2)),
 graph( 600, 600, -5, 10, -10, 10,x^(1/3),2^(2/5)*x^(1/5))) }}}