please help!! <3

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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!!

Guest Mar 17, 2020

geno3141 · Answer 1 · 2020-03-17T12:56:17

I'm assuming that you mistyped the problem:

4^x₁ = 5 ---> log(4^x₁) = log(5) ---> x₁ · log(4) = log(5) ---> x₁ = log(5) / log(4)

5^x₁ = 6 ---> log(5^x₁) = log(6) ---> x₁ · log(5) = log(6) ---> x₁ = log(6) / log(5)

6^x₁ = 7 ---> log(6^x₁) = log(7) ---> x₁ · log(6) = log(7) ---> x₁ = log(7) / log(6)

127^x₁ = 128 ---> log(127^x₁) = 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