Question

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`?

Answer 1

`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]