Question

prove that log 5 to the base 3 is irrational

Answer 1

let $log_35$= x/y
$3^ \frac{x}{y}$=5
$3^x=5^y$
since 3 and 5 are prime number, there are no integer after division. Hence, $log_35$ is irrational.

Answer 2

Answer:

$log_{3}5$  is irrational.

Step-by-step explanation:

Let us assume that $log_{3}5$ is rational, then it can be written in the form $3^{\frac{p}{q}}=5$, where p and q are both positive integers.

⇒$3^{p}=5^{q}$

Therefore, 3 must divide 5, but both 3 and 5 are co prime and 3 cannot divide 5, hence our assumption was wrong.

Therefore, $log_{3}5$ is irrational.