Suppose that #4^(x_1) = 5, 5^(x_2) = 6, 6^(x_3) = 7,...., 126^(x_123) = 127, 127^(x_124) = 128#. What is the value of the product #x_1x_2...x_124#?

1 Answer
Nov 27, 2016

Answer:

#3 1/2#

Explanation:

#4^(x_1)=5 #. Taking log of both sides we get #x_1log4=log5 or x_1=log5/log4#.
#5^(x_2)=6 #. Taking log of both sides we get #x_2 log5=log6 or x_2 =log6/log5#.
#6^(x_3)=7 #. Taking log of both sides we get #x_1log6=log7 or x_3=log7/log6#.
#..................#
#126^(x_123)=127 #. Taking log of both sides we get #x_123 log126=log127 or x_123=log127/log126#.
#127^(x_124)=128 #. Taking log of both sides we get #x_124 log127=log128 or x_124=log128/log127#.
#x_1*x_2*....*x124 = (cancellog5/log4) (cancellog6/cancellog5)(cancellog7/cancellog6)...log(cancel127/cancellog126)(log128/cancellog127) = log128/log4=log2^7/log2^2=(7cancellog2)/(2cancellog2)=7/2=3 1/2#[Ans]