If 3^a*4^b = c, what is the value of b? (1) 5^a = 25 (2) c = 36

Bunuel:
Pls correct my reasoning.

for Q1) my reasoning was since 3^a*4^b = c and base 3 is "even" and base 4 is "odd" then there wont be any other answer except a=2 and b=1. But I was wondering what are the other values of a and b can be for this equation to be true. Sorry, I am asking too much

Thanks for the awesome explanation. You rock Bunuel!

For 1: equation \(3^a*4^b = 36\) (if there is no restrictions on a and b) has infinitely many solutions for a and b: a=0 --> 4^b=36 --> b=log_4(36); a=1 --> 4^b=12 --> b=log_4(12), ...

Hope it's clear.

Thanks Bunuel. Makes sense, the power of logarithms! I was wondering whether the first question was 700+. This is a new perspective. I havent really used logs in any of questions. Awesome !

Very enlightening ! Thanks

Thanks for the explanation!

If you are wondering why stmnt 2 alone is not sufficient, think of it this way:

\(3^a4^b = c\)
(2) c = 36

So
\(3^a4^b = 36\)
Now for every value of a, there is a different value of b.
Say, a = 1, then 4^b = 12 and b = 1.79 approx
a = 2, then 4^b = 4 and b = 1
a = 3, then 4^b = 36/27 and b = 0.2 approx
and so on...

If we were given that a and b are integers, then answer would have been (B)

Dear Karishma Agreed with your approach i also think that way

but wanted to ask you one thing that does GMAT wants us to seriously think the nos to be real nos until specifically stated to be integers
or this kind of confusion is rare in real GMAT
or these kind of questions are actually TRAP questions

Sorry to say but yes, GMAT wants you to consider whether the number can be non-integer and if so, how your answer could vary. Also, GMAT is not above laying such and many other traps for you e.g. the easy C. This is one of the reasons why DS questions are considered much harder than PS questions. You have to analyze the problem from every aspect and consider every possibility. That is why it takes more time to do DS questions even though the concepts behind PS and DS questions are exactly the same. While practicing these questions, there will be many times when you will feel like pulling your hair out because you overlooked one tiny thing e.g. 'what if x is 0' or something like that and all effort was wasted because you got the answer wrong. Hence, try and be aware of these traps and practice as much as you can.

Karishma, Please tell me where exactly im going wrong

Statement 1)
a=2
Insufficient

Statement 2)
insufficient

Both taken together
b=1

You are not wrong. The answer is (C).
You need both statements to get the value of b.

If I solved the problem as:
3^a * 4^b = 36
3^a*2^2b = 3^2*2^2 [a = 2, b = 2]
What is the problem?

What is the difference between 3^a * 4^b = 36 and 5^21 x 4^11 =2x10^n

There is nothing wrong with the solution (a = 2, b = 1) except that it doesn't say that a and b are integers (as mentioned above) hence it is just one of the infinite solutions. a and b can be any real numbers and for every value of a, b will have a corresponding real value.

Three variables require three equations in normal scenario.
\(3^a*4^b = c\)



(1) \(5^a = 25……a=2\)
Nothing about b or c. b would depend on the value of c.
Insufficient

(2) \(c = 36\)
\(3^a*4^b = c=36\)
As a and b are not given to be integers, a can take any value and b will have a corresponding value that fits in.
Insufficient


Combined
\(3^a*4^b = c\)
\(3^2*4^b = 36…….b=1\)
Sufficient


C

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