Question 975062


{{{y= a*x^p}}} passes through the points ({{{3}}}, {{{2/81}}}) and ({{{6}}}, {{{1/1296}}})

Find the constants {{{a}}} and {{{p}}}.



{{{y= a*x^p}}} 
{{{log(y)= log(a*x^p )}}}
{{{log(y)= log(a)+log( x^p )}}}

{{{log(y)= log(a)+p*log( x )}}}

if given the points : 
({{{3}}},{{{ 2/81}}})=> {{{x=3}}} and {{{y=2/81}}}
 and 
({{{6}}}, {{{1/1296}}})=> {{{x=6}}} and {{{y=1/1296}}}

so, we use them to form a system of equations


{{{log(y)= log(a)+p*log( x )}}}  substitute {{{x=3}}} and {{{y=2/81}}}

{{{log(2/81)= log(a)+p*log( 3 )}}}  .....eq.1

{{{log(y)= log(a)+p*log( x )}}}   substitute {{{x=6}}} and {{{y=1/1296}}}

{{{log(1/1296)= log(a)+p*log( 6)}}}   .....eq.2



{{{log(2)-log(81)= log(a)+p*log( 3 )}}}  .....eq.1
{{{log(1)-log(1296)= log(a)+p*log( 6) }}}  .....eq.2..........write {{{81}}} as {{{3^4}}} and {{{1296}}} as {{{3^4*2^4}}}
--------------------------------------------------------------
{{{log(2)-log(3^4)= log(a)+p*log( 3 )}}}  .....eq.1
{{{log(1)-log(3^4*2^4)= log(a)+p*log( 2*3)}}}   .....eq.2
--------------------------------------------------------------------
{{{log(2)-4log(3)= log(a)+p*log( 3 )}}}  .....eq.1
{{{log(1)-(log(3^4)+log(2^4))= log(a)+p*(log( 2)+log(3))}}}   .....eq.2...substitute {{{log(1)=0}}}
--------------------------------------------------------------------------------
{{{log(2)-4log(3)= log(a)+p*log( 3 )}}}  .....eq.1
{{{0-4log(3)-4log(2))= log(a)+p*(log( 2)+log(3))}}}   .....eq.2
------------------------------------------------------------------subtract 1 from 2 

{{{-4log(3)-4log(2))-log(2)+4log(3)= log(a)+p*(log( 2)+log(3))-log(a)-p*log( 3 )}}}

{{{-5log(2)= p*(log( 2)+log(3)-p*log( 3 ))}}}

{{{-5log(2)= p*log( 2)}}}
 
{{{p=-5}}}

{{{log(2)-4log(3)= log(a)+p*log( 3 )}}}  .....eq.1...substitute {{{-5}}} for {{{p}}}

{{{log(2)-4log(3)= log(a)-5*log( 3 ) }}} 

{{{log(2)-4log(3)+5*log( 3 ) = log(a) }}}

{{{log(2)+log( 3 ) = log(a) }}}

{{{log(2*3)= log(a)}}} 

{{{log(6)= log(a) }}}

{{{a=6}}}

so, your function is: {{{y= 6x^(-5)}}} => {{{y= 6(1/x^5)}}} => {{{y= 6/x^5}}}