Problem 61 - College Algebra with Corequisite Support

Question

Is there a number \(x\) such that \(\ln x=2 ?\) If so, what is that number? Verify the result.

Step-by-Step Solution

Answer

Yes, the number is \( e^2 \).

Step 1: Understand the Problem Statement

We are asked to determine if there is a number \( x \) such that the natural logarithm of \( x \) equals 2, i.e., \( \ln x = 2 \).

Step 2: Apply the Definition of Logarithm

The natural logarithm \( \ln x \) is the power to which \( e \) (Euler's number, approximately 2.718) must be raised to get \( x \). Therefore, we need to find \( x \) such that \( e^2 = x \).

Step 3: Solve for x

From \( \ln x = 2 \), we apply the exponential form to find \( x \). Therefore, \( x = e^2 \).

Step 4: Verify the Result

To verify, substitute \( x = e^2 \) back into the equation: \( \ln(e^2) = 2 \cdot \ln e \). Since \( \ln e = 1 \), this simplifies to \( 2 \). Hence, \( x = e^2 \) is the correct solution.

Key Concepts

Exponential Equations

Exponential equations are a fascinating type of mathematical equations where variables appear in the exponent. In the equation \( \ln x = 2 \), we encountered the need to express a variable, \( x \), in terms of an exponential function. To solve such equations, you can use inverse operations. For example, the inverse operation of a logarithm is an exponentiation. This means if you have \( \ln x = 2 \), you solve for \( x \) by rewriting the equation in exponential form, known as exponentiating both sides. Thus, \( x = e^2 \), where \( e \) is the base of the natural logarithm. This step highlights the switch of roles between the base and the exponent, enabling solutions for values in different domains. Understanding this transformation is crucial for solving various real-world problems that involve growth processes or decay, modeled through exponential equations.

Euler's Number

Euler's number, denoted as \( e \), is a fundamental constant in mathematics, approximately equal to 2.71828. It is the base of the natural logarithm, one of the key pillars of continuous growth models in calculus. The curiosity about \( e \) arises from its unique properties, including the fact that the function \( e^x \) is its own derivative. This makes \( e \) vital not only for mathematics but also for fields like science, economics, and engineering that rely on modeling exponential growth or decay. When we solve our original problem \( \ln x = 2 \), we use \( e \) because the natural logarithm is precisely defined in relation to \( e \). The constant plays a pivotal role in transforming a logarithmic expression into an exponential one, as seen when \( ln x = 2 \) converts to \( x = e^2 \). In this exercise, Euler's number helps us comprehend how natural growth rates work and how to reverse a logarithmic expression back to its exponential form.

Verification of Solutions

Verification is an essential step in solving equations, ensuring that your solution holds true under the original conditions. When we find \( x = e^2 \) from \( \ln x = 2 \), it is important to check if this solution satisfies the given equation. We substitute \( x = e^2 \) back into the equation \( \ln(e^2) \). Using logarithmic properties, this expression evaluates as \( 2 \cdot \ln e \). Since \( \ln e = 1 \), the expression simplifies to 2. This confirms that our solution is correct. Verifying solutions involves re-evaluation through different equivalent expressions, providing confidence in the correctness and relevance of the solution. Keeping this phase in the solving process identifies errors and enriches understanding, crucial for mastering both textbook exercises and real-world problems.