MODERATE EXERCISES

(1) Prove: log₂ 5 is irrational.

Suppose log₂ 5 = a / b where a and b are integers. By the property of logs, 2^a/b = 5., which implies that 2^a = 5^b. Thus, 2^a = 5^b is an integer that is both even and odd. This is a contradiction which implies that log₂ 5 cannot be represented by the fraction a / b and is therefore irrational.