Question

prove that log 6 to the base 4 is irrational

Answer 1

Given:

log 6 to the base 4

To find:

Prove that log 6 to the base 4 is irrational

Solution:

From given, we have,

log 6 to the base 4

First, consider log_4 6 is rational (i.e. a quotient of integers)

log 6 to the base 4 = m/n

The m and n integers must be without common prime factors such that

4^ {m/n} = 6

We will show that m and n are both even

(4^{m/n})^n = 6^n

So

4^m = 6^n

Divide the base numbers by common factor, 2, so, we get,

2^m = 3^n

As the product of 2 odd numbers must be odd and the same applied for 2 even numbers, so, 2m and 3n are not equivalent and therefore m/n must not be rational.

Hence it is proved that log 6 to the base 4 is irrational