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There's this equation in chemistry called the hasselbalch equation. I looked at how it was derived, and it isn't making sense to me. $\ pKa -log(\frac{x}{y})$ equals to $\ pKa + log(\frac{y}{x})$, which confuses me because I thought the inverse would be $\ pKa -\frac{1}{log(\frac{x}{y})}$. Where did I go wrong, and could you please show me how to do it correctly.

This is my first time using MathJax (:

Redwood
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    This is general property of the logarithm function: $\log(\frac 1 x) = - \log(x)$. More generally, $\log(x^\alpha) = \alpha \log(x)$, and the previous case simply corresponds to $\alpha = -1$ – Numbra Jan 27 '21 at 07:38

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You're confusing additive and multiplicative inverses, and in any case you didn't change the overall sign, so all you've done is replace $a-b$ with $a-b^{-1}$ instead of $a+b^{-1}$ (which would still be wrong, but would at least result from confusing one "inverse" of $b$ with another). Note$$-\log\tfrac{x}{y}=-(\log x-\log y)=\log y-\log x=\log\tfrac{y}{x}.$$

J.G.
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