Question 1138426
<br>
{{{log((x^2y)) = n}}}
{{{2log((x))+log((y)) = n}}} (1)<br>
{{{log((x/y^2)) = p}}}
{{{log((x))-2log((y)) = p}}} (2)<br>
(1) and (2) are a pair of linear equation with log(x) and log(y) as the variables.  Solve the pair of equations for log(x) and log(y) in terms of n and p.<br>
{{{4log((x))+2log((y)) = 2n}}} (3)  [equation (1), multiplied by 2]
{{{5log((x)) = 2n+p}}}  [equation (2) plus equation (3)]
{{{log((x)) = (2n+p)/5}}}<br>
{{{-2log((x))+4log((y)) = -2p}}} (4)  [equation (2), multiplied by -2]
{{{5log((y)) = n-2p}}}  [equation (1) plus equation (4)]
{{{log((y)) = (n-2p)/5}}}<br>
Now we have expressions for log(x) and log(y) in terms of n and p, so we can find an expression for log(x/y) in terms of n and p.<br>
ANSWER:  {{{log((x/y)) = log((x))-log((y)) = ((2n+p)/5)-((n-2p)/5) = (3p+n)/5}}}<br>
CHECK:<br>
{{{(3p+n)/5 = (3(log((x))-2log((y)))+(2log((x))+log((y))))/5 = (5log((x))-5log((y)))/5 = log((x))-log((y)) = log((x/y))}}}