<s>[INST] Simplifying logarithm of a product My textbook reads (without explanation, naturally):
$\log \prod_{y=0}^{9}{(1+\frac{1}{10x+y})} = \log{(1+\frac{1}{x})}$
Wondering how this was achieved...
 [/INST] Notice that $1+\frac{1}{10x+y}=\frac{10x+y+1}{10x+y}$, so the product is: $$\prod_{y=0}^{9}{\left(1+\frac{1}{10x+y}\right)}=\frac{10x+1}{10x}\cdot\frac{10x+2}{10x+1}\cdot\frac{10x+3}{10x+2}\dots\frac{10x+10}{10x+9}.$$  Everything cancels out except the first denominator and the last numerator, giving $$\frac{10x+10}{10x}=\frac{x+1}{x}=1+\frac{1}{x}.$$
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