if log base a 3=2if a is irrational ?/give reason
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Answer:
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Step-by-step explanation:
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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.