Complicated Logarithm If x>0, y>0, and x^2+y^2=98xy then log(x+y) can be expressed as A log(x)+B log(y)+C where A,B,C are real numbers and all logarithms are base 10 logarithms. Compute 100ABC.

Maverick Avery

Maverick Avery

Answered question

2022-10-22

Complicated Logarithm
If x > 0, y > 0, and
x 2 + y 2 = 98 x y
then log ( x + y ) can be expressed as A log ( x ) + B log ( y ) + C where A , B , C are real numbers and all logarithms are base 10 logarithms. Compute 100 A B C.

Answer & Explanation

Pradellalo

Pradellalo

Beginner2022-10-23Added 16 answers

Here's a hint: Add 2 x y to both sides of your equation. Then recall that x 2 + 2 x y + y 2 = ( x + y ) 2 .
Full answer:
x 2 + 2 x y + y 2 = 100 x y ( x + y ) 2 = 100 x y 2 log ( x + y ) = log ( 100 ) + log ( x ) + log ( y ) 2 log ( x + y ) = log ( x ) + log ( y ) + 2 log ( x + y ) = 1 2 A log ( x ) + 1 2 B log ( y ) + 1 C 100 A B C = 100 1 2 1 2 1 = 25

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