Question

Problem 4. Prove that log, 11 is irrational (15 pts) Answer:

Answer 1

Answer and explanation:

To prove : $\log_2 11$ is irrational ?

Proof :

Let $\log_2 11$ is rational number.

So, It can be expressed in p/q form where, p and q are integers and q is non-zero.

$\log_2 11=\frac{p}{q}$

Using property of logarithm,

$11=(2)^{\frac{p}{q}}$

or $11^q=(2)^{p}$

Which means 11 must be divisible by 2 for some p and q,

But 11 and 2 are co-prime.

So, Our assumption is not true.

$\log_2 11$ is irrational number.

Hence proved.