Is the value of $\log_27$ a rational number?

Question

Is $\log_27$ a rational number?

Answer 1

Suppose $\log_2 7 = {a\over b}$ for two positive integers $a$ and $b$.

$\log_2 7 = { \ln 7 \over \ln 2} = {a \over b}$

Cross multiply,

$b \ln 7 = a \ln 2 \implies \ln ( 7^b ) = \ln (2^a)$

Take the $e^{( \ \ )}$ of both sides,

$7^b = 2^a$

This is impossible for integers $a$ and $b$ because $7^b$ is always going to be an odd number, while $2^a$ will always be an even number. They can never be equal, thus, $\log_2 7$ is not a rational number.

Answer 2

If $a,b$ are positive integers and $2^{a/b}=7$ then $2^a=7^b$ which make an even number equal to an odd number.