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1. if 4= 3, what is 42x-2 ?

2. Suppose that 4^a=5, 5^b = 6, 6^c=7, and 7^d=8. what is a * b * c * d ?

3. suppose that x^y = y^x and y=3/4x. what is x + y?

Dec 13, 2020

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1.  4x  = 3

42x - 2   =    42x / 42  =  (4x) 2/ (42)   =    (3)2  /16  =   9/16

2.   4a  = 5         5b=6       6c   =7    7d  = 8

If we  take  the log of  both sides  of the  first  thing we get

log 4a  = log 5     and we can write

a log 4  = log 5

a = log 5  / log 4

Following suite  we must have  that

b =log 6/log 5

c = log 7/ log 6

d = log 8 /log 7

So

a * b * c * d  =      (log 5/ log4)  (log6/log 5)  (log 7/ log 6)  ( log 8/ log 7) =  log 8 / log 4

3. suppose that x^y = y^x and y=3/4x. what is x + y?

x^y =  y^x         y = (3/4)x

x^[(3/4)x ]  = [(3/4)x ] ^x       take the log of  both sides

(3/4)x log x =  x log [ (3x / 4 ) ]       and the right side  becomes

(3/4)x  log x =  x  [ log 3 + log x  - log 4 ]

(3/4)xlog x  = x log 3  + x log x - x log 4

(3/4)xlog x - x log x =  x ( log 3 - log 4  )

(-1/4)x log x = x (log 3 - log 4)       divide  out  x

(-1/4)  log x  =  (log 3 - log 4)       multiply both sides  by -4

log x =   4 [ log 4 - log 3)]

log x  =  4 log ( 4/3)

log x  =  log (4/3)4

This implies that   x = (4/3)4  =  256 / 81

So  y= (3/4)x = (3/4) (256 / 81)   =  64 / 27  =  192 / 81

So

x +  y  =   [ 256 + 192 ]  /  81   =  448 / 81   Dec 13, 2020