Problem 94 - Precalculus : Building Concepts and Connections

Question

This set of exercises will draw on the ideas presented in this section and your general math background. Explain why In 4 is between 1 and \(2,\) without using a calculator.

Step-by-Step Solution

Answer

For any number \(x\) such that \(e < x < e^2\), the logarithm \(1 < \text{Ln}(x) < 2\). Since 4 falls between \(e\) and \(e^2\), \text{Ln}(4) must be between 1 and 2.

Step 1: Understand the nature of natural logarithm

It's important to note a few things about natural logarithms, often represented as Ln. In particular, for any number \(x\) such that \(0 < x < e\) (where \(e\) represents the base of the natural logarithm, approximately 2.71828), we know that \(0 < \text{Ln}(x) < 1\). Additionally, for any number \(x\) such that \(e < x < e^2\), we have \(1 < \text{Ln}(x) < 2\).

Step 2: Apply the natural logarithm rules

Taking the natural logarithm of 4, we know that 4 is greater than 1 but less than \(e^2\). Therefore, we can state confidently that the natural logarithm of 4 must be greater than 1 but less than 2.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms can simplify complex mathematical problems. Logarithms have unique properties due to their relationship with exponents:

• Product Property: When multiplying numbers, the logarithm of the product is the sum of the logarithms of each number. In terms of formula, \(\log_b (mn) = \log_b m + \log_b n\).This means if you have two numbers, say 2 and 3, and you want their logarithm in base 10, you can add the logarithm of 2 to the logarithm of 3.


• Quotient Property: For division, the logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator,\(\log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n\).This helps simplify logarithms of fractions.


• Power Rule: When raising a number to a power, the logarithm of that power is the exponent times the logarithm of the base number: \(\log_b (m^n) = n \cdot \log_b m\).

This property is useful in finding logarithms of numbers raised to powers. Like in your given exercise, knowing these properties helps in estimating where the natural logarithm value would fall without precise calculations.

Inequalities

Inequalities are tools that allow us to compare values that may not be known exactly.Understanding how to use inequalities can simplify many mathematical problems:

• Inequalities state that one value is larger or smaller than another, shown by signs such as ">" or "<".


• To make sense of these inequalities involving logarithms, it helps to know the range where the logarithmic function lies. In our exercise, since 4 is greater than 1 and less than \(e^2\), we conclude that\(1 < \ln(4) < 2\).

By comparing numbers within known intervals, we have a better understanding of their logarithmic values without direct computation. This application of inequalities is key to solving the exercise where you need to explain the logarithm's range.

Approximations

Approximations in mathematics involve estimating values that are difficult to compute exactly without advanced tools.They are particularly useful in problems involving logarithms where exact answers might not be readily available:

• The natural logarithm, denoted as \(\ln(x)\), is complex for numbers not close to simple multiples of the base, \(e\).For such numbers, we use approximations to get a sense of their value.


• Approximating helps to simplify mathematical operations, especially when details are not critically necessary.By knowing that 4 lies between 1 and \(e^2\), we approximate \(\ln(4)\) without needing to calculate it exactly.

This technique is helpful in this exercise to explain the value of \(\ln(4)\) without using any calculators. Approximations also solidify your understanding of underlying mathematical principles.

Mathematical Reasoning

Mathematical reasoning is the process of using logical thinking to deduce properties and establish truths within mathematics.It is the spine of problem-solving and involves understanding and connecting different mathematical concepts:

• Through reasoning, one can infer unknowns by utilizing known facts or properties.In the given exercise, you use the known properties of inequalities and exponential growth to reason out where \(\ln(4)\) exists.


• Effective reasoning often involves breaking down complex problems into smaller, more comprehensible parts, then connecting these simple conclusions back to the original query.

Using reasoning, we realize that since\(e < 4 < e^2\), \(\ln(4)\) logically falls between 1 and 2.This approach not only solves problems but also enhances your deep understanding of related mathematical concepts.