Question

If log 3 base a equal to 2, why a is irrational ? give reasons​

Answer 1

Answer:

Step-by-step explanation:

Answer 2

Step-by-step explanation:

log 3 base a = 2

=> a² = 3

=> a = √3

To prove : √3 is irrational

Assumption: Let √3 be rational.

So,

√3 = p/q {where q is not equals to 0, p and q have no common factor else than 1 and p and q are integers.}

√3 = p/q

=> 3 = p²/q²

=> 3q² = p²

As 3 divides 3q², so 3 divides p² but 3 is prime. This implies that 3 divides p.

Now,

Let p = 3m, where m is an integer.

Substituting this value of p, we get,

p² = 2q²

=> (3m)² = 3q²

=> 9m² = 3q²

=> 9m² = q²

As 3 divides 3m², so 3 divides q² but 3 is prime. This implies that 3 divides q.

Thus, from the above lines, we can see that p and q have a common factor 3. This contradicts that p and q have no common factor except 1.

Hence, it is proved that √3 is irrational.

As a = √3 , so , a is irrational number.

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