Problem 61 - College Algebra

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 \(e^2\) (approximately 7.389).

Step 1: Understand the Problem

The problem asks whether there is a number \(x\) such that \(\ln x = 2\). The natural logarithm function, denoted as \(\ln\), is the inverse of the exponential function \(e^x\). We need to find such an \(x\).

Step 2: Rewrite the Equation

Since the natural logarithm is the inverse of the exponential function, we can rewrite the equation \(\ln x = 2\) in terms of an exponential function as \(x = e^2\).

Step 3: Find the Number

Calculate the value of \(x\) by evaluating the expression \(x = e^2\). This gives an approximate value for \(x = 7.389\).

Step 4: Verify the Result

To verify, substitute \(x = e^2\) back into the original logarithmic equation: \(\ln(e^2) = 2\). Since \(\ln(e^2)\) simplifies to \(2\) by the property of logarithms, the solution is correct.

Key Concepts

Exponential Functions

Exponential functions are mathematical functions of the form \( f(x) = a^x \), where \( a \) is a positive constant called the base, and \( x \) is the exponent, which can be any real number. These functions are unique because they involve growth or decay at rates proportional to their current value. For example, the base \( e \) (approximately 2.718) is often used in mathematics, and its exponential function is written as \( e^x \). This is particularly important in continuous compounding, population models, and various natural phenomena.
In our exercise, the function \( e^x \) is used, translating the logarithmic equation \( \ln x = 2 \) into \( x = e^2 \). This transformation leverages the property that natural logarithms and exponential functions are inverses of each other.

Inverse Functions

Inverse functions essentially "undo" each other. If you have a function \( f(x) \), its inverse, denoted \( f^{-1}(x) \), will return the original value inputted into \( f(x) \). For instance, if \( y = f(x) \), then \( x = f^{-1}(y) \).
For exponential functions (\( e^x \)) and natural logarithms (\( \ln x \)), the inverse relationship is crucial. The statement \( \ln x = 2 \) means you want to find \( x \) that when plugged into \( \ln x \) gives \( 2 \). By using the inverse property, we find \( x \) by solving \( x = e^2 \). Hence, the exponential function is the inverse of the logarithmic function, allowing us to solve such equations.

Logarithmic Equations

Logarithmic equations involve logarithms, which are the inverse of exponential functions. Solving these equations often requires using logarithmic properties or changing the equation to an exponential form.
The natural logarithm, denoted \( \ln \), has \( e \) as its base. In mathematical terms, \( \ln \) answers the question, "To what power must \( e \) be raised, to obtain a certain number?"
In solving \( \ln x = 2 \), we rephrase the question to find the value of \( x \) that satisfies \( e^2 = x \). Thus, \( x = e^2 \) simplifies the solution. This step uses the fundamental property of logarithms and exponentials being inverse functions, which directly leads us to the solution.