\[log_23 \times log_3 4 \times log_4 5 \times ...log_{30}31 \times log_{31}32\]
?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
\[log_2 32=log_2 2^5= 5 log_22=5\]
.