Problem 53 - Algebra 2

Question

Solve each equation or inequality. Round to the nearest ten-thousandth. \(e^{5 x} \geq 25\)

Step-by-Step Solution

Answer

\(x \geq 0.6438\).

Step 1: Understanding the Equation

We need to solve the inequality \(e^{5x} \geq 25\). This means we are looking for values of \(x\) that will make the expression \(e^{5x}\) greater than or equal to 25.

Step 2: Take the Natural Logarithm

To isolate the variable \(x\), we need to take the natural logarithm of both sides of the inequality. This gives us: \[ \ln(e^{5x}) \geq \ln(25) \] Using the property of logarithms \(\ln(e^y) = y\), this simplifies to:\[ 5x \geq \ln(25) \]

Step 3: Solve for x

Now that we have simplified the inequality to \(5x \geq \ln(25)\), the next step is to solve for \(x\) by dividing both sides by 5:\[ x \geq \frac{\ln(25)}{5} \]

Step 4: Calculate \(\ln(25)\)

Calculate \(\ln(25)\) using a calculator to get a numerical value. The natural logarithm of 25 is approximately 3.2189.

Step 5: Compute the Division

Now that we know \(\ln(25) \approx 3.2189\), we divide this by 5:\[ x \geq \frac{3.2189}{5} \] This simplifies to approximately \(x \geq 0.6438\).

Step 6: Round to the Nearest Ten-Thousandth

The calculated value is already accurate to the ten-thousandth place, so the final result, \(x \geq 0.6438\), is already in the required rounded format.

Key Concepts

Understanding Inequalities

Inequalities are mathematical statements that show the relationship between expressions that are not equal. When dealing with inequalities, we use symbols like \(<\), \(>\), \(\leq\), and \(\geq\). These symbols help us understand the comparative relationship between quantities.
When solving inequalities, we aim to find all possible values of the variable that makes the inequality true. This involves:

• Analyzing the inequality to understand what is being asked
• Performing mathematical operations to isolate the variable
• Paying attention to the direction of the inequality, which might change in certain situations like multiplying or dividing by a negative number

In our example, the inequality \(e^{5x} \geq 25\) tells us to find values of \(x\) where \(e^{5x}\) is at least 25. This process requires a methodical approach to handle the inequality appropriately.

Role of Natural Logarithms in Solving Inequalities

Natural logarithms, denoted as \(\ln\), play a key role in solving inequalities, especially when they involve exponential expressions. The natural logarithm is the inverse function of the exponential function with base \(e\), where \(e\) is approximately 2.71828.
In the context of the inequality \(e^{5x} \geq 25\), applying the natural logarithm helps us simplify the expression. Here's how it works:

• We take the \(\ln\) of both sides, which leads to \(\ln(e^{5x}) \geq \ln(25)\).
• Utilizing the property \(\ln(e^y) = y\), we simplify this to \(5x \geq \ln(25)\).

Understanding this process is crucial, as it transforms a complex exponential inequality into a simpler linear form, making it easier to solve. Once simplified, we use arithmetic operations to isolate \(x\).

Exploring Exponential Functions

Exponential functions are a class of mathematical functions that involve powers of a constant base. In the most common scenario, the base is \(e\), the natural exponential constant, making the function \(e^x\) a standard form.
Exponential functions grow rapidly and have unique properties:

• They are continuous and always positive.
• They grow faster than polynomial functions for large values of \(x\).
• They have the inverse function known as the natural logarithm.

These characteristics are important, particularly when solving inequalities like \(e^{5x} \geq 25\). Here, recognizing that \(e^{5x}\) represents rapid growth helps us understand why logarithms are used to solve it. Logarithms "undo" the exponential, simplifying the inequality for further manipulation. Through the natural logarithm's property, we're able to isolate and determine the boundary values required for \(x\). Such insights aid significantly in tackling complex problems involving exponential expressions.