If #theta# goes from #theta_1# to #theta_2#, then the arc length is #sqrt{2}(e^{theta_2}-e^{theta_1})#.
Let us look at some details.
#L=int_{theta_1}^{theta_2}sqrt{r^2+({dr}/{d theta})^2}d theta#
since #r=e^{theta}# and #{dr}/{d theta}=e^{theta}#,
#=int_{theta_1}^{theta_2}sqrt{(e^{theta})^2+(e^{theta})^2}d theta#
by pulling #e^{theta}# out of the square-root,
#=int_{theta_1}^{theta_2}e^{theta}sqrt{2} d theta=sqrt{2}int_{theta_1}^{theta_2}e^{theta} d theta#
by evaluating the integral,
#=sqrt{2}[e^{theta}]_{theta_1}^{theta_2}=sqrt{2}(e^{theta_2}-e^{theta_1})#
