1. Identify the type of salt

The salt $BA$ is formed from:

• weak acid: $HA$
• weak base: $BOH$

So, $BA$ is a salt of a weak acid and weak base.

For such a salt, the pH is given by:

$\mathrm{pH} = 7 + \frac{1}{2}\log\left(\frac{K_b}{K_a}\right)$

This can also be written as:

$\mathrm{pH} = 7 + \frac{1}{2}(\mathrm{p}K_a - \mathrm{p}K_b)$

because

$\log\left(\frac{K_b}{K_a}\right)=\log K_b - \log K_a = -\mathrm{p}K_b + \mathrm{p}K_a$

2. Substitute the given values

Given:

$\mathrm{p}K_a = 4.80, \qquad \mathrm{p}K_b = 4.78$

So,

$\mathrm{pH} = 7 + \frac{1}{2}(4.80 - 4.78)$

$\mathrm{pH} = 7 + \frac{1}{2}(0.02)$

$\mathrm{pH} = 7 + 0.01 = 7.01$

3. Match with the options

The calculated pH is:

$\boxed{7.01}$

So the correct option is:

C: 7.01

4. Compare with stored correct answer

Stored correct answer: C

Our derived answer: C

They match.