Solve 2^x=13mod 3^4

Amiya Melendez 2022-10-15 Answered
Solve 2 x = 13 mod 3 4
I know log 13 = 30 mod 3 4 and log 16 = 15 mod 3 4
I've tried subbing log 13 / log 16 for 2 but I am not sure what to do next.
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Answers (2)

giosgi5
Answered 2022-10-16 Author has 15 answers
(1) 2 x = 13 mod 3 4
means that 2 x = 3 4 k + 13 = 81 k + 13 for some k Z
For 81 k + 13 to equal 2 x , for some k , x Z , k must be odd ( 81 k + 13 must be even).
For k = 1, it exists no x Z such that 81 + 13 = 2 x
But for k = 3, you get 81 3 + 13 = 256 = 2 8 = 2 x
Hence, x = 8 is one (integer) solution of (1).
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hogwartsxhoe5t
Answered 2022-10-17 Author has 3 answers
In this case you're looking for log 2 13 ( mod 3 4 ), and you can write
log 2 13 = log 13 log 2 ( mod 3 4 )
Moreover, we know log 16 = log 2 4 = 4 log 2. So log 2 = 1 4 log 16. Substituting this yields
log 2 13 = log 13 1 4 log 16 = 4 log 13 log 16 ( mod 3 4 )
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