Problem 54 - Algebra 2

Question

Solve each equation or inequality. Round to the nearest ten-thousandth. \(e^{-2 x} \leq 7\)

Step-by-Step Solution

Answer

\( x \geq -0.9729 \)

Step 1: Isolate the Exponential Expression

The given inequality is: \[ e^{-2x} \leq 7 \]There is no need to isolate the exponential further since it already stands alone on one side of the inequality.

Step 2: Apply the Natural Logarithm

To solve the inequality involving an exponential function, apply the natural logarithm to both sides to eliminate the exponential expression:\[ \ln(e^{-2x}) \leq \ln(7) \] This simplifies to: \[ -2x \leq \ln(7) \] since \( \ln(e^y) = y \).

Step 3: Solve for x

Now, solve for \(x\) by dividing both sides by \(-2\). Remember to reverse the inequality sign when dividing by a negative:\[ x \geq -\frac{\ln(7)}{2} \]

Step 4: Calculate the Value

Compute the value of \(-\frac{\ln(7)}{2}\) using a calculator:\[ \ln(7) \approx 1.9459 \]So,\[ x \geq -\frac{1.9459}{2} \approx -0.97295 \]Round this to the nearest ten-thousandth:\[ x \geq -0.9729 \]

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a special type of logarithm that has the mathematical constant \( e \) as its base. The constant \( e \), approximately equal to 2.71828, is a fundamental irrational number in mathematics. The natural logarithm is widely used because it simplifies calculations involving exponentials.
For instance, when you have an equation of the form \( e^x = a \), taking the natural logarithm of both sides yields \( x = \ln(a) \). This property stems from the definition of logarithms: the logarithm is the exponent to which the base must be raised to obtain a specific value.
Key points about the natural logarithm include:

• \( \ln(1) = 0 \) because any number to the power of 0 is 1.
• \( \ln(e) = 1 \) since \( e^1 = e \).
• Natural logs are the inverses of exponential functions with base \( e \).

Understanding how to manipulate natural logarithms is crucial when solving exponential equations and inequalities, as it allows for the exponent to be expressed in simpler, linear terms.

Inequalities

Inequalities involve mathematical expressions where one side is not necessarily equal to the other, and they use symbols like \( \leq \), \( \geq \), \( < \), and \( > \). In the context of solving inequalities, the solution indicates a set of values that satisfy the condition instead of a single value. This makes inequalities a fundamental concept in algebra and many other math applications.
When working with inequalities:

• The direction of the inequality sign matters. It shows the relationship between two expressions.
• When both sides of an inequality are multiplied or divided by a negative number, the inequality sign reverses direction. This is crucial when handling negative coefficients.
• Simplifying inequalities often involves similar steps as solving equations, ensuring that all terms are properly managed on one side.

Understanding inequalities in exponential contexts means being able to convert exponential relationships into linear ones using logarithms, making them simpler to analyze and solve.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They are usually represented as \( f(x) = a^x \) and are fundamental in describing growth and decay processes. A common base for exponential functions is the number \( e \), which is used in continuous growth scenarios.
In exponential inequalities, such as \( e^{-2x} \leq 7 \), the goal is to find all possible values of \( x \) that satisfy the inequality. Exponential functions possess some distinctive features:

• The base \( a \) is a constant positive number, typically greater than 1 for growth, or between 0 and 1 for decay.
• They are continuously increasing or decreasing, depending on their form.
• Utilizing the natural logarithm can simplify inequalities involving exponential functions, converting them into manageable linear expressions.

By understanding exponential functions and how they interact with logarithms, one can effectively tackle a range of problems involving exponential growth, decay, and inequalities.