I agree that converting from base 5 to base 10 and then from base 10 to base 2 is easiest, although the conversion can be done directly.
Here are alternatives for the methods for doing both conversions, without using the place values in base 5 and base 2.
(1) Converting 134 base 5 to base 10.
The method is to start with the leftmost digit and repeatedly multiply by the base and add the next digit, until all digits have been used:
1*5 = 5; 5+3 = 8
8*5 = 40; 40+4 = 44
134 base 5 = 44 base 10
(2) Converting 44 base 10 to base 2.
The method is to repeatedly divide by the new base, 2, noting the remainders each time. Those remainders are the base 2 digits, starting with the rightmost digit:
44/2 = 22 remainder 0
22/2 = 11 remainder 0
11/2 = 5 remainder 1
5/2 = 2 remainder 1
2/2 = 1 remainder 0
1/2 = 0 remainder 1
44 base 10 = 101100 base 2
