What are the products of real solutions of this equation? How can I solve log_(1/2)^2(4x)+ log_2((x^2)/8)=8 ? I have tried the elementary for logarithms simplifying the terms in brackets.

Juan Lowe

Juan Lowe

Answered question

2022-11-08

What are the products of real solutions of this equation?
How can I solve log 1 / 2 2 ( 4 x ) + log 2 ( x 2 8 ) = 8 ?
I have tried the elementary for logarithms simplifying the terms in brackets.

Answer & Explanation

Stella Andrade

Stella Andrade

Beginner2022-11-09Added 19 answers

Let log 2 x = t. Then
log 1 / 2 ( 4 x ) = log 2 ( 4 x ) log 2 ( 1 / 2 ) = ( 2 + t )
log 2 x 2 8 = 2 t 3
So we solve ( 2 + t ) 2 + 2 t 3 = 8 ( t 1 ) ( t + 7 ) = 0, or x = 2 , 1 2 7 for a product of 1 64
spasiocuo43

spasiocuo43

Beginner2022-11-10Added 6 answers

Going to natural logarithms (the only I know, if I may confess), you have
log 1 / 2 ( 4 x ) = log ( 4 x ) log ( 2 ) = 2 log ( x ) log ( 2 )
log 2 ( x 2 8 ) = log ( x 2 8 ) log ( 2 ) = 2 log ( x ) log ( 2 ) 3
So, setting t = log ( x ) log ( 2 ) = log 2 ( x ),
log 1 / 2 2 ( 4 x ) + log 2 ( x 2 8 ) = ( 2 + t ) 2 + 2 t 3
and after development, the equation to solve is then
t 2 + 6 t 7 = ( t 1 ) ( t + 7 ) = 0

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