The correct answer to your question is that you must recognize the kind of solution you have. Sometimes that is easier said than done, especially until you get some practice at it. Practice makes perfect.
1. If it is a salt (a,b,d), the Bronsted-Lowry Theory tells you. In the case of NaNO3, neither Na+ nor NO3- is hydrolyzed; therefore, NaNO3 is neutral in water solution. For b (potassium benzoate), the benzoate ion is hydrolyzed, the K+ is not, so the benzoate takes the H+ and releases OH- and the solution is basic. (But see below for what I use.)
2. For solution that are not salts but are mixtures, such as c, e, and f, use of the Henderson-Hasselbalch equation will tell you. (But see below to see how I do it.)
Frankly, for salts, I use another approach. It will be frowned on by your teacher, I'm sure, but it works every time. For NaNO3,
NaNO3 + HOH ==> NaOH + HNO3.
The reaction actually does not take place BUT you can use it to answer the question of acidity or basicity. NaOH is a strong base. HNO3 is a strong acid. Therefore, the salt is neutral in water. What about KC6H5CO2?
C6H5COOK + HOH ==> C6H5COOH + KOH
C6H5COOH is a weak acid. You can find it listed in Ka tables. KOH is a strong base. In water, solution, the salt is basic (because of the strong base part). etc. I will leave d for you.
For c, the H-H equation is
pH = pKa + log (base/acid).
pH = look up the value of Ka (and convert to pKa) or pKa C6H5COOH and substitute for pKa in the equation. Then substitute into the log part. A 50:50 mixture means base and acid or equal so base/acid = 1 and log 1 = 0; therefore, pH = pKa and you will know acid or base for the solution.
Be sure to follow up if some of this is murky.
What do I do for mixtures? If it is a mixture of weak acid and its salt the solution will be acidic. If it is a mixture of a weak base and its salt, the solution will be basic.
Thank you so much!