Given that log_2(3) • log_3(4) • log_4(5)..... log_n-1(n) • log_n(n+1) = 10, find the value of n.

Question

Given that log_2(3) • log_3(4) • log_4(5)..... log_n-1(n) • log_n(n+1) = 10, find the value of n.

 

NEED ANSWER ASAP

Answer 1

Okay, so this question is pretty dope because it is telescoping, so here it goes.

 

One wants to start with how you would casually solve any logarithm problem and take both sides as a power. So we start with

 

log₂(3)log₃(4)log₄(5)..... log_n-1(n)log_n(n+1) = 10

 

Since our first base is 2, take both sides to be powers of 2 as below:

 

2^{10 =}2^{log₂(3)log₃(4)log₄(5)..... log_n-1(n)log_n(n+1)} 

 

=3^{log₃(4)log₄(5)..... log_n-1(n)log_n(n+1)}

 

=...

 

=n^log_n(n+1)

 

=n+1

 

So we end up with n=2¹⁰-1. The ellipses in the middle is just all the terms cancelling out (telescoping) as in the second equality. I really do hope you appreciate this problem for how pretty it is, please let me know if you need anything cleared up.