Solving the system with logarithms I tried solving the system {((4x)^(log_2(2y))=64),(8y)^(log_2(2y))=256:} several times but still keep getting wrong solutions.

janine83fz

janine83fz

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2022-08-19

Solving the system with logarithms
I tried solving the system { ( 4 x ) log 2 ( 2 y ) = 64 ( 8 y ) log 2 ( 2 y ) = 256 several times but still keep getting wrong solutions.

Answer & Explanation

Bryant Liu

Bryant Liu

Beginner2022-08-20Added 15 answers

log 2 4 x log 2 2 y = log 2 64 = 6
log 2 2 y log 2 8 y = log 2 256 = 8
Some simplifications leads to:
( log 2 4 + log 2 x ) ( log 2 2 + log 2 y ) = 6
( log 2 2 + log 2 y ) ( log 2 8 + log 2 y ) = 8
Call u = log 2 x, v = log 2 y
( 2 + u ) ( 1 + v ) = 6
( 1 + v ) ( 3 + v ) = 8
The solutions are:
{ u = 1 , v = 1 } { u = 7 2 v = 5 }
Then:
{ x = 2 , y = 2 } { x = 2 7 2 , y = 2 5 }

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