Question

why is log 2 irrational

Answer 1

Answer:

Step-by-step explanation:

Short proof of “log 2 is irrational”

 

               Assume that log 2 is rational, that is,

                                        (1)

               where p, q are integers.

               Since log 1 = 0 and log 10 = 1,  0 < log 2 < 1  and therefore p < q.

               From (1),          

                                               

                                                  , where q – p is an integer greater than 0.

               Now, it can be seen that the L.H.S. is even and the R.H.S. is odd.

               Hence there is contradiction and  log 2  is irrational.

Answer 2

Assume that log 2 is rational, that is,

(1)

where p, q are integers.

Since log 1 = 0 and log 10 = 1, 0 < log 2 < 1 and therefore p < q.

From (1),

, where q – p is an integer greater than 0.

Now, it can be seen that the L.H.S. is even and the R.H.S. is odd.

Hence there is contradiction and log 2 is irrational.