Question 18

$$\log_{5}{2}$$ is

Solution

Let $$\log_{5}{2}$$ = y

Let us assume $$\log_{5}{2}$$ is a rational number.

$$\log_{5}{2}$$ = p/q, where p and q are co primes.

5^(p/q)=2 => 5^p=2^q.

5^p=5*5*5*5*5*5*5..................p times

2^p=2*2*2*2*2*2*2..................q times

No value of p and q can satisfy the equation. Hence y is an irrational number.

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