Could someone explain to me why the negative part in the second term switches
$\ln (1+x) - \ln (1-x)$ becomes $\ln \frac{1+x}{x-1}$.
I understand the division just not why $-x$ becomes $+x$ and the $+1$ becomes $-1$.
Many thanks.
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3Assuming you mean $\ln(\frac{1+x}{x-1})$ on the right, it's not true. Try $x = 0$, for instance. – Alex Provost Feb 08 '23 at 14:36
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What's the context in which this came up? – eyeballfrog Feb 08 '23 at 14:37
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Do you mean $\ln|1+x|-\ln|1-x|=\ln\left|\frac{1+x}{x-1}\right|$? If so, I'm guessing you forgot the mod signs when a logarithm comes out of integration. (Before anyone else mentions it, there's one further subtlety to such integration on $\Bbb R$.) – J.G. Feb 08 '23 at 14:40