Question

Is (1) log 2 rational or irrational? Justify your answer.
(ii) log 100 rational or irrational? Justify your answer​

Answer 1

Answer:

1. Assume that log 2 is rational, that is,

$log(2) = \frac{p}{q}$

where p, q are integers.

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

From (1),

$2 = {10}^{ \frac{p}{q} }$

${2}^{q - p} = {5}^{p}$

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 and log 2 is irrational.

1. log100

Let log10 (100) = x

log10 (10)^2 =x

⇒10^x =10^2

∴x=2

∴log100 is rational.