1. if 4x = 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?
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