Consider the dissociation of a weak acid, HA:
HA rightleftharpoons H^+ + A^-. As we know,
K_a = {[H^+][A^-]}/[[HA]],
and log_10{K_a} = log_10[H^+] + log_10{{[A^-]]/[[HA]]};
Equivalently (multiplying each by -1,
-log_10{ K_a} = -log_10[H^+] - log_10{{[A^-]]/[[HA]]}.
But, by definition, -log_10{ K_a} = pK_a, and -log_10[H^+] = pH.
Therefore, pH = pK_a + log_10{{[A^-]]/[[HA]]}. This is a form of the buffer equation, with which you are going to get very familiar.
Now at half-equivalence, by definition, [HA] = [A^-], and since log_10 1 = 0, when plugged back into the equation, pH = pK_a. So, in order to measure pK_a values of weak acids we plot a titration curve with a pH meter, and note value at half-equivalence.
