SOLUTION: log4 y + logy 4 =5/2

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Question 459621: log4 y + logy 4 =5/2
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
log%284%2C%28y%29%29+%2B+log%28y%2C%284%29%29%22%22=%22%225%2F2
Use the identity:

log%28A%2C%28B%29%29%22%22=%22%221%2Flog%28B%2C%28A%29%29, 0%3CA%3C%3E1, 0%3CB%3C%3E1 

on the second term:

log%284%2C%28y%29%29+%2B+1%2Flog%284%2C%28y%29%29%22%22=%22%225%2F2

Let U=log%284%2C%28y%29%29

U%2B1%2FU=5%2F2

Clear of fractions by multiplying through by 2U

2U%5E2+%2B+2+=+5U

Get 0 on the right:

2U%5E2-5U%2B2=0 

Factor the left side:

%282U-1%29%28U-2%29=0

2U-1=0
2U=1
U=1%2F2

U-2=0
U=2

For U=1%2F2
Replace U by U=log%284%2C%28y%29%29

log%284%2Cy%29=1%2F2

Replace by the exponential equivalent:

y=4%5E%281%2F2%29=sqrt%284%29=2

For U=2
Replace U by U=log%284%2C%28y%29%29

log%284%2Cy%29=2

Replace by the exponential equivalent:

y=4%5E2=16

So there are two solutions, y = 2 and y = 16.

Edwin