Question

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

just do it.​

Answer 1

Step-by-step explanation:

∵ log₂5 € Real.

Let, log₂5 be a rational number.

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

→$5 = 2^{ \frac{p}{q} }$

(taking 'q' both side)

$5^q = 2^q5 ....1$

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

Hence the contradiction we suppose is wrong .

Hence , log₂5 is an irrational number.

Answer 2

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 ] .

$5^q = 2^q$

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.