SOLUTION: If p,q>0, log base p of q+ log base qof p=5/2 and pq=54√ 2, then find (p+q)/3

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Question 1197722: If p,q>0, log base p of q+ log base qof p=5/2 and pq=54√ 2, then find (p+q)/3
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


(1)

Given: log%28p%2Cq%29%2Blog%28q%2Cp%29=5%2F2

log%28p%2Cq%29 and log%28q%2Cp%29 are reciprocals: log%28p%2Cq%29%2B1%2Flog%28p%2Cq%29=5%2F2

Solve A+1/A = 5/2 to find the addends are 2 and 1/2: log%28p%2Cq%29=2

log%28q%29%2Flog%28p%29=2

log%28q%29=2%2Alog%28p%29

log%28q%29=log%28p%5E2%29

q=p%5E2

(2)

pq=54sqrt%282%29

Substitute q=p%5E2 and simplify:

p%28p%5E2%29=54sqrt%282%29

p%5E3=54sqrt%282%29

p=3sqrt%282%29

q=p%5E2=18

(3) Evaluate the given expression:

%28p%2Bq%29%2F3=%283sqrt%282%29%2B18%29%2F3=sqrt%282%29%2B6

ANSWER: 6%2Bsqrt%282%29

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Added in response to question from student....

I arbitrarily chose to let log%28p%2Cq%29=2. If I had chosen to let log%28p%2Cq%29=1%2F2, the result would have been the same, with p and q switched.