The #pH# and #pOH# scales were developed in the days before electronic calculators as a means to facilitate chemical calculations. And they used a logarithmic scale, the which chemists, and engineers, would have to look up, and manipulate.
In aqueous solution under standard conditions, we often speak of #K_w#, the ion product...where...
#K_w=[H_3O^+][HO^-]=10^(-14)#...
And we can take #log_10# of both sides to give....
#log_(10)K_w=log_(10)10^(-14)=log_10[H_3O^+]+log_10[HO^-]#.
And thus.... #-14=log_(10)[H_3O^+]+log_(10)[HO^-]#
Or.....
#14=-log_(10)[H_3O^+]-log_(10)[HO^-]#
#14=underbrace(-log_10[H^+])_(pH)underbrace(-log_10[OH^-])_(pOH)#
#14=pH+pOH#
By definition, #-log_10[H^+]=pH#, #-log_10[HO^-]=pOH#
