In aqueous solution an acid #HA# undergoes the following equilibrium....
#HA(aq) + H_2O(l) rightleftharpoonsH_3O^+ + A^-#
Clearly, #K_a=([H_3O^+][A^-])/([HA(aq)])#
And for the reverse reaction...
#A^(-) + H_2O(l) rightleftharpoonsHA(aq) + HO^-#
Clearly, #K_b=([HA(aq)][HO^-])/([A^-])#
And #K_axxK_b=([H_3O^+][A^-])/([HA(aq)])
*([HA(aq)][HO^-])/([A^-])#
#=[H_3O^+][HO^-]=K_w=10^-14#
And should we take #-log_10# of BOTH sides, we gets....
#-log_10[H_3O^+]-log_10[HO^-]=-log_10(10^-14)=14#
#underbrace(-log_10[H_3O^+])_(pH)underbrace(-log_10[HO^-])_(pOH)=14#
And so #pOH + pH=14#
And likewise we could take...#-log_10# of #K_axxK_b# to get....
#pK_a+pK_b=14#