Exponential And Logarithmic Functions

To prove that the function is a bijection, we need to show that it is both injective (one-to-one) and surjective (onto). 1. Injectivity: - For a function to be injective, if , then must hold. - Let , i.e., - By the property of logarithms, we can rewrite this as Therefore, the function is injective. 2. Surjectivity: - A function is surjective if for every , there exists such that . - Let be arbitrary. - We need to find such that - This is equivalent to which is always a positive real number for any (since and ). Therefore, is surjective. Since the function is both injective and surjective, it is bijective. %Quuciktip