Question

Is (1) log 2 rational or irrational? Justify your answer.​

Answer 1

Answer: Log 2 is irrational.

Answer 2

Answer: Log 2 is irrational

Step-by-step explanation:

 Short proof of “log 2 is irrational” :-

 Assume that log 2 is rational, that is, log 2 = p / q .......... (i.)

 where p, q are integers.

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

From (i.), 2 = 10 to the power p / q

= 2 to the power q = ( 2 * 5 ) to the power p.

= 2 to the power q - p = 5 to the power p , 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.

Hope it helps :)