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Chapter 17: Problem 38

Solve the inequalities by displaying the solutions on a calculator. $$\ln (x-3) \geq 1$$

Short Answer

Expert verified
The solution is \( x \geq 5.71828 \).

Step by step solution

01

Understand the Inequality

The inequality is given as \( \ln (x-3) \geq 1 \). This means we want to find values of \( x \) such that the natural logarithm of \( x-3 \) is greater than or equal to 1.
02

Solve the Basic Inequality

To solve \( \ln(x-3) = 1 \), we need to exponentiate both sides to remove the natural logarithm. Doing so yields: \[ e^{\ln(x-3)} = e^1 \] Therefore, \( x-3 = e \), so \( x = e + 3 \).
03

Determine the Solution Set

The inequality \( \ln(x-3) \geq 1 \) implies \( x-3 \geq e \). Since \( e \approx 2.718 \), adding 3 gives \( x \geq 3 + e \).
04

Interpret on a Calculator

On a calculator, calculate \( e \) to several decimal places (\( e \approx 2.71828 \)) and add 3. Therefore, \( x \geq 5.71828 \) to at least five decimal places.
05

State the Solution

The solution to the inequality is all real numbers \( x \) such that \( x \geq 3 + e \). Therefore, the solution is \( x \geq 5.71828 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Logarithm
The natural logarithm, represented as \( \ln \), is a logarithm to the base \( e \), where \( e \) is approximately equal to 2.71828. It is a mathematical function often used to describe exponential growth or decay processes. In simple terms, the natural logarithm answers the question: "To what power must \( e \) be raised, to obtain a given number?" For instance, if \( \ln(a) = b \), it means \( e^b = a \). This comes in handy when solving equations or inequalities involving logarithmic expressions.
  • Key feature: \( \ln(e) = 1 \), because \( e^1 = e \).
  • Useful property: \( \ln(xy) = \ln(x) + \ln(y) \).
These properties make \( \ln \) an essential tool in algebra.
Exponentiation
Exponentiation refers to the operation of raising a number (the base) to a certain power (the exponent). In the context of natural logarithms, exponentiation is used to "undo" the logarithm. When you have an equation like \( \ln(x-3) = 1 \), you can exponentiate both sides to eliminate the logarithm and solve for \( x \).
  • Example: \( e^{\ln(x-3)} = e^1 \), simplifies to \( x-3 = e \).
  • The operation is based on properties such as \( a^0 = 1 \) and \( a^1 = a \).
Knowing how to use exponentiation is critical in converting between logarithmic and exponential equations.
Inequality Solution
Solving an inequality involves finding all possible values of the variable that satisfy the given condition. In our exercise, the inequality was \( \ln (x-3) \geq 1 \). To solve it:1. First, find the critical point by equating \( \ln(x-3) = 1 \), leading to the point \( x = e + 3 \).2. Then, determine the range of \( x \) that satisfies \( \ln(x-3) \geq 1 \). This translates to \( x \geq e + 3 \), meaning \( x \) must be greater than or equal to this value.3. Using a calculator further refines this to \( x \geq 5.71828 \).Calculators can help verify or explore these solutions by providing accurate values for \( e \) and calculating the values to several decimal places.
Mathematical Solutions
Mathematical solutions often require multiple steps and techniques to arrive at a precise answer. With the inequality \( \ln(x-3) \geq 1 \), the analytical approach involves:- Understanding the role of each function, here \( \ln \) and exponentiation.- Calculating crucial constants like \( e \) to an adequate precision.- Verifying on a calculator can provide extra confidence, especially when dealing with transcendental numbers like \( e \).This solution process shows how different mathematical concepts align to provide a comprehensive understanding. Moreover, it's crucial to interpret results contextually, ensuring they fit within the given parameters of the problem.

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Most popular questions from this chapter

Draw a sketch of the graph of the given inequality. $$y \leq 15-3 x$$

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