Problem 2 - Basic Analysis: Introduction to Real Analysis

Question

Let \(b>0, b \neq 1\) be given. a) Show that for every \(y>0,\) there exists a unique number \(x\) such that \(y=b^{x} .\) Define the logarithm base \(b\), \(\log _{b}:(0, \infty) \rightarrow \mathbb{R},\) by \(\log _{b}(y):=x\) b) Show that \(\log _{b}(x)=\frac{\ln (x)}{\ln (b)}\). c) Prove that if \(c>0, c \neq 1,\) then \(\log _{b}(x)=\frac{\log _{c}(x)}{\log _{c}(b)}\) d) Prove \(\log _{b}(x y)=\log _{b}(x)+\log _{b}(y),\) and \(\log _{b}\left(x^{y}\right)=y \log _{b}(x)\).

Step-by-Step Solution

Answer

The logarithm base \( b \) is defined as the unique \( x \) such that \( y = b^x \), with properties such as \( \log_b(x) = \frac{\ln(x)}{\ln(b)} \) and \( \log_b(xy) = \log_b(x) + \log_b(y) \).

Step 1: Define the Logarithm

To define the logarithm base \( b \), we start with the function \( y = b^x \). For a given \( y > 0 \), there must be an exponent \( x \) such that \( y = b^x \), because the exponential function \( b^x \) is continuous and strictly increasing when \( b > 1 \) and continuous and strictly decreasing when \( 0 < b < 1 \). This guarantees that for each \( y > 0 \), we can find such a unique \( x \), which we define as \( x = \log_b(y) \).

Step 2: Show the Relationship with Natural Logarithms

To demonstrate that \( \log_b(x) = \frac{\ln(x)}{\ln(b)} \), recall the change of base formula. Starting with \( y = b^x \), taking natural logarithms on both sides gives \( \ln(y) = \ln(b^x) = x \ln(b) \). Solving for \( x \) gives \( x = \frac{\ln(y)}{\ln(b)} \). Thus, \( \log_b(x) = \frac{\ln(x)}{\ln(b)} \).

Step 3: Prove the Change of Base Formula

Use the relationship from Step 2 to show \( \log_b(x) = \frac{\log_c(x)}{\log_c(b)} \). Since \( \log_c(x) = \frac{\ln(x)}{\ln(c)} \) and \( \log_c(b) = \frac{\ln(b)}{\ln(c)} \), it follows that \( \log_b(x) = \frac{\frac{\ln(x)}{\ln(c)}}{\frac{\ln(b)}{\ln(c)}} = \frac{\ln(x)}{\ln(b)} \), confirming the equivalence with the previous expression.

Step 4: Prove the Properties of Logarithms

To show \( \log_b(xy) = \log_b(x) + \log_b(y) \), consider \( z = xy \). Then \( z = b^a \cdot b^b = b^{(a+b)} \). Therefore, \( \log_b(z) = a+b = \log_b(x) + \log_b(y) \). For \( \log_b(x^y) = y \log_b(x) \), if \( x^y = b^a \), then \( a = y \cdot \log_b(x) \), confirming the property.

Key Concepts

Exponential Functions

Exponential functions are fundamental in many areas of mathematics and science. At their core, these functions take the form \( y = b^x \), where \( b \) is a positive real number greater than zero and not equal to one. The variable \( x \) is the exponent that determines the power to which the base \( b \) is raised.

There are a few key traits of exponential functions:

• Continuity: Exponential functions are continuous, meaning they do not have any breaks or holes in their graphs.
• Monotonicity: If the base \( b > 1 \), the function is strictly increasing. If \( 0 < b < 1 \), the function is strictly decreasing.
• Unique Solutions: Given any \( y > 0 \), there exists a unique solution \( x \) such that \( y = b^x \). This property is crucial in defining logarithms.

Understanding these characteristics helps us define the logarithm base \( b \), which is essentially the inverse operation to the exponential function. For a given \( y \), the logarithm tells us the exponent \( x \) such that \( b^x = y \). This ability to "reverse" the process of exponentiation is what makes logarithms valuable.

Change of Base Formula

The Change of Base Formula is a powerful tool in simplifying logarithmic expressions and solving logarithmic equations. It provides a way to express a logarithm in terms of logs with another base, typically the natural logarithm (base \( e \)).

The formula is expressed as:

• \( \log_b(x) = \frac{\ln(x)}{\ln(b)} \)

Here's why this formula works:

• Start with \( y = b^x \) and take the natural logarithm of both sides: \( \ln(y) = x \ln(b) \).
• Solve for \( x \): \( x = \frac{\ln(y)}{\ln(b)} \).
• This means \( \log_b(x) = \frac{\ln(x)}{\ln(b)} \), simplifying the calculation of any base logarithm to one involving just base \( e \) logarithms.

Additionally, this concept extends beyond natural logarithms:

• For any base \( c > 0, c eq 1 \), the formula can be generalized: \( \log_b(x) = \frac{\log_c(x)}{\log_c(b)} \).

This generalization makes it versatile for numerous mathematical applications, providing consistency regardless of the base you start with.

Properties of Logarithms

Logarithms are governed by certain rules that make manipulating them more straightforward, especially when dealing with multiplication, division, and powers. These properties stem directly from the properties of exponents, given their inverse relationship to logarithms.

Key properties include:

• Product Property: \( \log_b(xy) = \log_b(x) + \log_b(y) \). This property states that the log of a product is equal to the sum of the logs of its factors. It originates from exponent rules, as multiplying like bases adds exponents.
• Quotient Property: Although not shown explicitly in the original exercise, it's useful to note: \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \). This mirrors the division of exponents.
• Power Property: \( \log_b(x^y) = y \log_b(x) \). This property demonstrates how an exponent can be "brought down" to multiply the logarithm, similar to multiplying exponents being the same as raising a power to another power.

These properties are not only convenient for solving logarithmic equations but also provide insights into the behavior of exponential growth and decay in various real-world scenarios. Mastering them is key to understanding logarithms on a deeper level.