Question 1197722
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(1)<br>
Given: {{{log(p,q)+log(q,p)=5/2}}}<br>
{{{log(p,q)}}} and {{{log(q,p)}}} are reciprocals: {{{log(p,q)+1/log(p,q)=5/2}}}<br>
Solve A+1/A = 5/2 to find the addends are 2 and 1/2: {{{log(p,q)=2}}}<br>
{{{log(q)/log(p)=2}}}<br>
{{{log(q)=2*log(p)}}}<br>
{{{log(q)=log(p^2)}}}<br>
{{{q=p^2}}}<br>
(2)<br>
{{{pq=54sqrt(2)}}}<br>
Substitute {{{q=p^2}}} and simplify:<br>
{{{p(p^2)=54sqrt(2)}}}<br>
{{{p^3=54sqrt(2)}}}<br>
{{{p=3sqrt(2)}}}<br>
{{{q=p^2=18}}}<br>
(3) Evaluate the given expression:<br>
{{{(p+q)/3=(3sqrt(2)+18)/3=sqrt(2)+6}}}<br>
ANSWER: {{{6+sqrt(2)}}}<br>
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Added in response to question from student....<br>
I arbitrarily chose to let {{{log(p,q)=2}}}.  If I had chosen to let {{{log(p,q)=1/2}}}, the result would have been the same, with p and q switched.<br>