Hydronium and hydroxide ions are present both in pure water and in all aqueous solutions. The concentrations of these ions in a solution are important in determining the solution’s properties and the chemical behaviors of its other solutes, and specific vocabulary has been developed to describe these concentrations in relative terms.
A solution is neutral if it contains equal concentrations of hydronium and hydroxide ions; acidic if it contains a greater concentration of hydronium ions than hydroxide ions; and basic if it contains a lesser concentration of hydronium ions than hydroxide ions.
Commonly used concentrations of hydronium and hydroxide are usually small and can span many orders of magnitude (i.e. many powers of 10 in their concentrations: solutions in lab commonly range from $10^{-6}$ to $10^{+1}$ M). Talking about small concentrations as decimal values can be confusing (imagine comparing 0.0000023 vs 0.000023 M frequently). So we often use a logarithmic scale – the p-scale – to express concentrations.
For a generic quantity “X”, the pX (read “p of X”),is defined with a base-10 logarithm:
$$pX=-logX$$The pH of a solution is therefore: $$pH=-log[H_3O^+]$$ where [H3O+] is the molar concentration of hydronium ion in the solution.
Rearranging this equation to isolate the hydronium ion molarity yields the equivalent expression:
$$[H_3O^+]=10^{-pH}$$Likewise, the hydroxide ion molarity may be expressed as a p-function, or pOH:
$$pOH=-log[OH^-]$$or
$$[OH^-]=10^{-pOH}$$