SOLUTION: If log^x base 2 + log^² base x = (5/2), find all the value of x

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Question 1202796: If log^x base 2 + log^² base x = (5/2), find all the value of x
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The use of "^" with logarithms is not standard, and rather confusing....

In words, the equation is "log base 2 of x, plus log base x of 2, equals 5/2".

log%282%2Cx%29%2Blog%28x%2C2%29=5%2F2

log%282%2Cx%29 and log%28x%2C2%29 are reciprocals.

The sum, 5/2, is 2 + 1/2; and those are reciprocals.

So we have either...

(1)
log%282%2Cx%29=2
x=2%5E2=4
log%28x%2C2%29=log%284%2C2%29=1%2F2

or

(2)
log%282%2Cx%29=1%2F2
x=2%5E%281%2F2%29=sqrt%282%29
log%28x%2C2%29=log%28sqrt%282%29%2C2%29=2

ANSWERS:
x=4
x=sqrt%282%29