Two possible solutions are:

a=1, b=1 ,c=1         &

a=4,b=2,c=2

So, abc = 1 or 16.

there are only 2 intgral sol a=1,b=1,c=1 and a=4,b=2,c=2.

As we know that if there is 4^a+3^b=5^c then it is a pythagorean triplet and pythagorean triplets have only one integral sol that is 2.

Also graphically if we see then these graphs are like the graph of exponential function  which does not have other integral values than the above given values.

hope u got it.

There is one more I can see:

a = 2

b = 0 

c = 1

I am trying to get to the solution. I think it is something to do with binomial theorem

i have found all the +ve integral soln. and as oneyeartogo said that a=2,b=0,c=1 is also a soln is correct but i dont think now there r any more soln to this.