1.Find (log_2 x)^2 if log_2 (log_8 x)=log_8(log_2 x) 2. If x,y >0, log_y x+log_x y=10/3 and xy =144, then find (x+y)/2

atestiguoki

atestiguoki

Answered question

2022-08-06

1.Find ( log 2 x ) 2 if log 2 ( log 8 x ) = log 8 ( log 2 ) x.
2.If x , y > 0 , log y x + log x y = 10 3 and xy=144, then find x + y 2

Answer & Explanation

Angel Burton

Angel Burton

Beginner2022-08-07Added 12 answers

1.
As log 2 ( log 8 x ) = log 8 ( log 2 x )
log 2 ( log 8 x ) = log 2 ( log 2 x ) log 2 8 ( R e c a l l : log a b = log e b log e a )
log 2 ( log 8 x ) = log 2 ( log 2 x ) log 2 2 3 = log 2 ( log 2 x ) 3
3 log 2 ( log 8 x ) = log 2 ( log 2 x )
log 2 ( log 8 x ) 3 = log 2 ( log 2 x )
( log 8 x ) 3 = log 2 x (Change of base once again)
( log 2 x ) 2 = ( log 2 8 ) 3 = 3 3 = 27 (We already seen before why log 2 8 = 3)
2.As log y x + log x y = 10 3
1 log x y + log x y = 10 3
Let u = log x y
1 u + u = 10 3
3 u 2 10 u + 3 = 0
( 3 u 1 ) ( u 3 ) = 0
u = 1 3 or u=3
If log x y = 1 3 x 1 3 = y
As x y = 144 y = 144 x
x 4 3 = 144 x = 41.57 y = 3.46
x + y 2 = 22.515
If log x y = 3 x 3 = y = 144 x
x 4 = 144 x = 3.46 y = 41.57

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