Question

${\log }_{3}5$ is

• A. a rational number
• B. an irrational number
• C. an integer
• D. a fraction

Answer 1

Answer: (B)

an irrational number

Let's assume ${\log }_{3}5$ is rational number,

${\log }_{3}5=\frac{p}{q},p>q>1$
As ${\log }_{3}5>1$ and non-integer.
Then $5=3^{p/q}$ or $5^q=3^p$

Which is not possible
$\because 5$ divides $5^q$ but $5$ cannot divide $3^P$ for any $p$.

$\therefore 5^q\ne 3^p$, which is a contradiction, so ${\log }_{3}5$ is an irrational number.