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Chapter 6: Problem 49

Solve the inequality.\(\ln x \geq 3\)

Short Answer

Expert verified
The solution to the inequality is \( 0 < x \leq e^3 \)

Step by step solution

01

Convert the Logarithmic Inequality to Exponential Form

In order to convert the inequality from logarithmic to exponential form, recall that \(\ln a = b \) is equivalent to \( e^b = a \). So \(\ln x \geq 3 \) is equivalent to \( e^3 \geq x \)
02

Evaluate \( e^3 \)

Calculate the value of \( e^3 \) using a calculator or known value of e (approximately 2.71828).
03

Write the Final Solution

After calculating \( e^3 \), the inequality should read \( x \leq e^3 \). The inequality solution is the set of x that satisfies this inequality, and since the logarithm function is only defined for positive x, the solution is \( 0 < x \leq e^3 \).

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Key Concepts

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

Exponential Functions
Exponential functions are math expressions where a constant base is raised to a variable exponent. The most common base in applications is the constant 'e'. This is often encountered in natural logarithms since it provides a natural rate of growth. In general, an exponential function can be written as:
  • If \( y = b^x \), where \( b \) is a constant and \( x \) is a variable, then \( y \) changes proportionally to its own value.
  • The base 'e' is approximately 2.71828 and is derived from values that come up in continuous growth phenomena.
Such functions are key in modeling growth processes, physics phenomena, and even in financial calculations. Understanding that the natural logarithm and exponential functions are inverse operations can help solve many real-world problems.
Natural Logarithm
The natural logarithm, denoted as \( \ln(x) \), is the logarithm to the base \( e \). It's a special logarithmic function because of its natural occurrence in mathematics, particularly in calculus.
  • One key property to remember is that \( \ln(e) = 1 \) because \( e^1 = e \).
  • This logarithm is very useful because it transforms multiplicative relationships into additive ones, simplifying many complex calculations.
  • In solving inequalities like \( \ln x \geq 3 \), the natural logarithm helps translate logarithmic expressions into exponential ones, making them more straightforward to solve.
It's crucial to understand how natural logs work, especially because they simplify certain types of equations that might otherwise be too complex to handle.
Inequality Solving
Solving inequalities often involves understanding both the properties of inequalities and the functions involved. With logarithmic inequalities like \( \ln x \geq 3 \), the approach involves:
  • Converting the logarithmic inequality into its exponential form. This is achieved by understanding the relationship \( \ln a = b \) being equivalent to \( e^b = a \).
  • Evaluating the expressions accurately, such as calculating \( e^3 \) using approximate values.
  • Simplifying the equation to identify the set of possible solutions. Remember that natural logs are only defined for positive values of \( x \), guiding the solution to \( 0 < x \leq e^3 \).
By breaking down each step, inequality solving becomes manageable with practice and comprehension of basic principles.

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Most popular questions from this chapter

Tell whether the function \(f(x)=\frac{1}{3} e^{4 x}\) represents exponential growth or exponential decay. Explain.

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