Question

Show that log ( base 2) 5 is irrational number.​

Answer 1

Step-by-step explanation:

∵ log₂5 € Real.

Let, if possible log₂5 is a rational number.

∵ log₂5 = p/q, and p, q € I and q ≠ 0 .

→ 5 = $2^{ \frac{p}{q} }$ .

[ take power 'q' both side ] .

$5^q = 2^q$ .....1 .

Which is Not possible, because equation 1 only right when q = 0 , but q ≠ 0 .

Therefore, our contradiction i.e.,log₂5 is a rational number is wrong .

Therefore, log₂5 is an irrational number.

Hence, it is proved.