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## Homework Statement

Prove that [itex] log_2(3) [/itex] is irrational.

## The Attempt at a Solution

This is also equivalent to [itex] 2^x=3 [/itex] from the definition of logs.

Proof: For the sake of contradiction lets assume that x is rational and that their exists integers P and Q such that x=P/Q .

so now we have [itex] 2^{\frac{P}{Q}}=3 [/itex]

now I will take both sides to the Q power .

so now we have [itex] 2^P=3^Q [/itex]

since P and Q are integers, there is no possible way to have 2 raised to an integer to equal 3 raised to an integer, because 2^P will always be even and 3^Q will always be odd. so this is a contradiction and therefore x is irrational.