Data Sufficiency

I think (1) is 5^a = 25.
Solution:
It is given that 3^a * 4^b = c.
Consider first (1) alone.
5^a = 25.
So a = 2.
So 3^2 * 4^b = c.
Or 4^b = c/(3^2).
To know b, we need to know c.
So (1) alone is not sufficient.
Next consider (2) alone.
c= 36.
So 3^a * 4^b = 36 = 2^2 * 3^2.
Or 2^2b * 3^a = 2^2 * 3^2.
We can equate a = 2 and 2b = 2 only if we were given that a and b are integers. Otherwise it is not necessary.
For example suppose a = b.
So we get that 3^a * 4^a = 36.
Or 12^a = 36.
Or a = ln36/ln12 which is not an integer.
So (2) alone is not sufficient.
Next combine both the statements together and check.
So we get a = 2.
Or 4^b = 36/(3^2) = 36/9 = 4.
Or b = 1.
So both (1) and (2) together are sufficient.

The correct answer is (C) .

Let b = a.
We get that when a = b, 12^a = 36.
Taking log on both sides, we have that a * log 12 = log 36.
Or a = (log 36)/(log 12).
So we can say that b = a = log 36/ log 12 (you will have to refer to log tables to get exact value).
Now next let b = 2a.
So 3^a * 4^2a = 36.
Or 3^a * 16^a = 36.
Or (48)^a = 36.
So a = log 36/ log 48.
So b = 2a = 2 * log 36/log 48.
So we can get different values of b from (2) alone.
So (2) alone is not sufficient.

Hope it is clearer!