I understand subtraction negates addition, and since log and exponents are basically addition and subtraction on steroids, then log must be the way to negate exponents. So from this rationale, is it fair to say that log is negative because the exponents are negative? That if it were any other quantities that involved whole numbers, then log would be positive?
For context, the equation for pH is:
$$[\text{H}^+][\text{OH}^-]=1×10^{-14} \text{ mol}^{2} \text{ dm}^{-6}$$
And if you know the number of hydronium ions present, then it can be found with the equation:
pH = – log(H+)
, which is the equation referred to in the question.
Best Answer
Under reasonable conditions, at any given moment, only a tiny fraction of water molecules are dissociated into the ${\rm H}^+$ and ${\rm OH}^{-}$ ions. Therefore, the molar concentrations of these ions are very small numbers, so their logarithms are negative. The customary definition of pH takes this into account.
If we add acid to water, then ${\rm H}^+$ ions' concentration grows by orders of magnitude, but still there are way fewer ${\rm H}^+$ ions than water molecules, and the concentration of ${\rm H}^+$ ions still remains a small number. So the logarithm of ${\rm H}^+$ ions' concentration remains negative in the vast majority of situations, and the pH definition still works as intended, stripping the minus sign of the logarithm.