(log x)^2 +(log y)^2-log(xy) log (x/y)
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Answer:
Consider the given equation.
2log(x
2
y)=3+log x−log y
log (x
2
y)
2
=3+log
y
x
log (x
4
y
2
)−log
y
x
=3
log (x
4
y
2
×
x
y
)=3
log (xy)
3
=3
(xy)
3
=10
3
...........(log
a
b
=x⇒b=a
x
)
xy=10⇒y=
x
10
........ (1)
And (x−y)=3........(2)
substitute equation (1) in (2)
x−
x
10
=3
x
x
2
−10
=3
x
2
−3x−10=0
x
2
−5x+2x−10=0
x(x−5)+2(x−5)=0
(x−5)(x+2)=0
x=5 or −2
But x>0
Hence, x=5 is the answer.
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