The pKb of ammonia, NH3, is 4.74 at 25°C.  Which of the following is the Ka of the ammonium ion, NH4+?

The easy way.

pKa + pKb = pKw = 14
You have pKa of 9.26 which is right but you took the -log of 4.26 which isn't right.
pKa = 9.26 = -log Ka
Ka = ?

thank you silly of me and to think i spent so long on that on a silly error.

Ahhh---but we do that sometimes. It is a GREAT learning experience. We aren't likely to do it twice.:-)

To find the Ka of the ammonium ion (NH4+), you can use the relationship between Ka and Kb.

First, you correctly calculated the pKa by subtracting the pKb value (4.74) from 14. You obtained a pKa value of 9.26.

Next, to find the Ka value, you need to convert the pKa value to its antilogarithm.

The formula to convert the negative logarithm to the value itself is:
antilogarithm = 10^(-pKa)

So, to find Ka, you can use the following calculation:
Ka = 10^(-pKa)

Substituting the pKa value (9.26) into the equation, you'll get:
Ka = 10^(-9.26)

Calculating this expression will give you the value of Ka for the ammonium ion.