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2^a + 2^b + 2^c = 148 .Find the value of a,b,and c.

Conventional method . All the three terms are some power of two  When x (=a,b,c)  = 1,2,3.4.5,6,7 , 2^x = 2,4,8,16,32,64,128. .The sum of three numbers among these numbers must  equal to 148. ie. 148= , 128 + 16 + 4.i.e., a=2,b.=4 and c = 7

Alternative method.  2^a + 2^b + 2^c =   2^a [ 1 + (2^b + 2^c)/ 2^a ]= 148. That is 148 is divisible by 2^a

2^a [ 1 + (2^b + 2^c)/ 2^a ]= 2² x 37; a = 2 and 2^b + 2^c = 144 ; 2^b [1+ 2^c-b] = 2⁴ x 9 ; b = 4; 2^c-b = 8 or c = 7

Other examples 

3^a + 3^b + 3^c = 759;    a = 1, b = 3, c = 6

If the base numbers are different method-II will not be suitable as the factor for one of the terms is the factor for the sum of the two other terms . e.g.,  2^a + 3^b + 4^c = 531;    a = 5, b = 5, c = 4 and 2^a + 3^b + 5^c = 300;    a = 5, b = 5, c = 2