Is log 2 a rational or an irrational? Justify your answer.

1.	Is log 2 a rational or an irrational? Justify your answer.
Answer» Assume that log 2 is rational, that is, log2=p/q(1) where p, q are integers. Since log 1 = 0 and log 10 = 1,0 < log 2 < 1and therefore p < q. From (1),2= 10^(p/q)2^q=(2×5)^p2(p-q)=5^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 andlog 2is irrational. Like if you find it useful

Answer»

Assume that log 2 is rational, that is,

log2=p/q(1)

where p, q are integers.

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

From (1),2= 10^(p/q)2^q=(2×5)^p2(p-q)=5^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 andlog 2is irrational.

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