If 2*log_e(x−2y)=log_ey+log_ex, then find the numerical value of x/y

capellitad9

capellitad9

Answered question

2022-07-22

If 2 l o g e ( x 2 y ) = l o g e y + l o g e x , then find the numerical value of x y
My try:
2 l o g e ( x 2 y ) = l o g e y + l o g e x
l o g e ( x 2 y ) 2 = l o g e x y
( x 2 y ) 2 = x y
But expanding the LHS doesn't give the result. What should I do further?

Answer & Explanation

nuramaaji2000fh

nuramaaji2000fh

Beginner2022-07-23Added 18 answers

Hint: Let z = x y , dividing your last equation by y 2 (note that y > 0, as log y is defined), we get
x y = ( x y 2 ) 2 z = ( z 2 ) 2
Now solve for z
Jaylene Hunter

Jaylene Hunter

Beginner2022-07-24Added 2 answers

Hint :
2 ln ( x 2 y ) = ln y + ln x ln ( x 2 y ) ln y = ln x ln ( x 2 y )
i.e.,
log ( x 2 y y ) = log ( x x 2 y )
i.e.,
x y 2 = 1 1 2 ( y x )
Can you complete this...?

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