On page 46 of Rouse Ball and HSM Coxeter's Mathematical Recreations and Essays, 13th ed., appears this step, in reference to the birthday problem $$ (1-\frac{1}{365})(1-\frac{2}{365})...(1-\frac{n-1}{365})=1/2 $$ "by taking logarithms, we obtain approximately" $$ \frac{1}{365}+\frac{2}{365}+...+\frac{n-1}{365}=log_e 2 $$
I'm unfamiliar with the step taken, and after some thinking and searching, haven't found any justification for the step.
They use the linear approximation $$ \log(1-x) \approx -x $$ which is good for $x$ close to $0$. And of course $\log(xy) = \log x + \log y$.