Fix b>1, y>0, and prove that there is a unique real x such...

Fix $b>1, y>0$, and prove that there is a unique real $x$ such that $b^{x}=y$, by completing the following outline. (This $x$ is called the logarithm of $y$ to the base b.) (a) For any positive integer $n, b^{n}-1 \geq n(b-1)$. (b) Hence $b-1 \geq n\left(b^{1 / n}-1\right)$. (c) If $t>1$ and $n>(b-1) /(t-1)$, then $b^{1 / n}<t .$ (d) If $w$ is such that $b^{w}<y$, then $b^{w+(1 / n)}<y$ for sufficiently large $n$; to see this, apply part $(c)$ with $t=y \cdot b^{-w}$ (e) If $b^{w}>y$, then $b^{w-(1 / m)}>y$ for sufficiently large $n$. (f) Let $A$ be the set of all $w$ such that $b^{w}<y$, and show that $x=\sup A$ satisfies $b^{x}=y .$ (g) Prove that this $x$ is unique.

Fix $b>1, y>0$, and prove that there is a unique real $x$ such that $b^{x}=y$, by completing the following outline. (This $x$ is called the logarithm of $y$ to the base b.)
(a) For any positive integer $n, b^{n}-1 \geq n(b-1)$.
(b) Hence $b-1 \geq n\left(b^{1 / n}-1\right)$.
(c) If $t>1$ and $n>(b-1) /(t-1)$, then $b^{1 / n}<t .$
(d) If $w$ is such that $b^{w}<y$, then $b^{w+(1 / n)}<y$ for sufficiently large $n$; to see this, apply part $(c)$ with $t=y \cdot b^{-w}$
(e) If $b^{w}>y$, then $b^{w-(1 / m)}>y$ for sufficiently large $n$.
(f) Let $A$ be the set of all $w$ such that $b^{w}<y$, and show that $x=\sup A$ satisfies $b^{x}=y .$
(g) Prove that this $x$ is unique.

Key Concepts

Inequalities and Induction Methods

Inequalities and induction are key tools used in establishing properties of exponential functions. For example, demonstrating that b^n - 1 is at least n(b - 1) for positive integers n requires an inductive argument. Such inequalities help in estimating the behavior of exponential functions, proving their continuity, and refining bounds. These estimates become essential when proving continuity and limits, thus underpinning arguments that lead to the determination and uniqueness of the logarithm.

Monotonicity

Monotonicity refers to the property of a function being either entirely non-decreasing or non-increasing over its domain. In the context of exponential functions, when the base b is greater than 1, the function is strictly increasing. This strict monotonicity is essential because it guarantees that each output value corresponds to exactly one input value, which makes the function invertible. As a result, the logarithm, as the inverse function, is well-defined and unique for each positive y.

Completeness and the Supremum Property

The completeness of the real numbers is a foundational property that states every nonempty set of real numbers that is bounded above has a least upper bound, or supremum. This property is critical in the proof of the existence of logarithms; by considering the set of all exponents that yield values below a given y, one can define the logarithm as the supremum of this set. The completeness property ensures that this supremum is a real number and, with additional arguments, that it is the unique solution of the equation b^x = y.

Exponential Functions

Exponential functions are mappings of the form f(x) = b^x, where the base b is a fixed positive real number greater than 1. These functions are characterized by their rapid growth, continuous behavior, and strict monotonicity, meaning they are either strictly increasing or strictly decreasing. Understanding their behavior is fundamental in various areas of mathematics because they model continuous growth and decay processes, and their properties facilitate the construction of inverse functions such as logarithms.

Logarithm as the Inverse Function

The logarithm function is defined as the inverse of the exponential function. For a given positive base b (with b > 1) and a positive number y, the logarithm log_b(y) is the unique real number x satisfying b^x = y. This inverse relationship is crucial because it allows us to solve exponential equations and is widely applied in areas like algebra, calculus, and complex analysis. The existence and uniqueness of the logarithm depend fundamentally on the properties of exponential functions, particularly their monotonicity.

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