Problem 34 - Precalculus

Question

In Exercises \(34-39,\) solve the inequality analytically. $$ e^{x}>53 $$

Step-by-Step Solution

Answer

For the inequality \( e^x > 53 \), \( x > 3.9703 \).

Step 1: Understand the Inequality

We need to solve the inequality \( e^x > 53 \). This involves finding the values of \( x \) that satisfy this condition. \( e^x \) is an exponential function, which means it's increasing for all real values of \( x \).

Step 2: Take the Natural Logarithm

To isolate \( x \), we take the natural logarithm on both sides of the inequality. The inequality becomes \( \ln(e^x) > \ln(53) \).

Step 3: Simplify Using Logarithm Properties

Using the property \( \ln(e^x) = x \), the inequality simplifies to \( x > \ln(53) \).

Step 4: Compute the Natural Logarithm

Use a calculator to find \( \ln(53) \). This gives us \( \ln(53) \approx 3.9703 \).

Step 5: Write the Solution Set

The solution to the inequality is \( x > 3.9703 \). We can express this solution as \( x \) belongs to the interval \( (3.9703, \infty) \).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. The general form is \( a^x \), where \( a \) is the base and \( x \) is the exponent. Exponential functions are powerful because they model situations involving rapid growth or decay, such as population growth or radioactive decay. The base \( e \) is particularly important, approximately equal to 2.718, and is used often in natural exponential functions, like \( e^x \). These functions are one-to-one and always increase as the exponent \( x \) becomes larger. This behavior is crucial in understanding how to solve inequalities involving exponential expressions.When solving an inequality, like \( e^x > 53 \), we are looking for all the values of \( x \) that make this inequality true. In this context, knowing that \( e^x \) is an increasing function tells us there will be a specific point where \( e^x \) becomes larger than 53, and for all \( x \) values greater than that point, the inequality will hold true.

Natural Logarithms

The natural logarithm, denoted as \( \ln \), is the logarithm to the base \( e \). It is the inverse function of the natural exponential function \( e^x \). This means that \( \ln(e^x) = x \) and \( e^{\ln(x)} = x \). In solving inequalities involving exponentials, like \( e^x > 53 \), the natural logarithm is used to "bring down" the exponent. By applying \( \ln \) to both sides, the inequality \( e^x > 53 \) becomes \( \ln(e^x) > \ln(53) \). This transformation leverages the property that allows \( \ln(e^x) \) to simplify to \( x \). It helps us transition from working with the exponential equation to a more manageable linear form, \( x > \ln(53) \). This step is foundational because it changes the problem into a simpler inequality, allowing for straightforward evaluation of the solution set through the computation of \( \ln(53) \).

Properties of Logarithms

Logarithms have several important properties that simplify the process of solving equations and inequalities.

• One of the key properties is the power rule: \( \ln(a^b) = b \cdot \ln(a) \). This is particularly useful when dealing with exponential expressions.
• Another important property is \( \ln(e^x) = x \), which stems from the fact that \( \ln \) and \( e^x \) are inverse functions. This property allows us to simplify expressions and inequalities involving \( e^x \).
• The property \( \ln(ab) = \ln(a) + \ln(b) \) and \( \ln(a/b) = \ln(a) - \ln(b) \) provide means to handle logarithms of products and quotients effectively.

These properties are crucial in solving inequalities like \( e^x > 53 \). They enable the transformation of the exponential inequality to a linear one, thus facilitating easier numerical computation and interpretation of the solution.

Solution Sets

Solution sets in inequalities describe the range of values that satisfy the inequality. For \( e^x > 53 \), we found that \( x > \ln(53) \). After calculating \( \ln(53) \approx 3.9703 \), our solution set becomes \( x > 3.9703 \). This can be expressed in interval notation: \( (3.9703, \infty) \), which means that any value greater than 3.9703 is a solution.

• Interval notation helps portray a range instead of individual numbers, representing infinite solutions effectively.
• This approach simplifies visualizing inequalities, illustrating that the solution continues indefinitely past the boundary point.
• Understanding solution sets can aid in fields requiring optimization, such as economics or engineering.

By clearly defining and notating solution sets, we form a complete understanding of all possible solutions, key in contexts where criteria must be met or surpassed.