Suppose that \(4^{x_1}=5, 5^{x_2}=6, 6^{x_3}=7, \dots, 127^{x_{124}}=128. \) What is \(x_1x_2\cdots x_{124}\)??
Please help!!
I'm assuming that you mistyped the problem:
4x1 = 5 ---> log(4x1) = log(5) ---> x1 · log(4) = log(5) ---> x1 = log(5) / log(4)
5x1 = 6 ---> log(5x1) = log(6) ---> x1 · log(5) = log(6) ---> x1 = log(6) / log(5)
6x1 = 7 ---> log(6x1) = log(7) ---> x1 · log(6) = log(7) ---> x1 = log(7) / log(6)
127x1 = 128 ---> log(127x1) = log(128) ---> x1 · log(127) = log(128)
---> x1 = log(128) / log(127)
[ log(5) / log(4) ] · [ log(6) / log(5) ] · [ log(7) / log(6) ] · ... · [ log(128) / log(127) ] = log(128) / log(4)
= log(27) / log(22) = [ 7 · log(2) ] / [ 2 · log(2) ] = 7/2