## A Telescoping Logarithmic Series

What is the value of
$log_23 \times log_3 4 \times log_4 5 \times ...log_{30}31 \times log_{31}32$
?
Working from the right hand side, we can write
$log_{30}31 \times log_{31}32= log_{30}31^{log_{31}32}=log_{30}32$

$log_{29}30 \times log_{30}31 \times log_{31}32= log_{29}30 \times log_{30}32=log_{29}30^{log_{30}32}=log_{29}32$

Continuing in this we can rewrite the expression as
$log_2 32=log_2 2^5= 5 log_22=5$
.