Problem 47 - Algebra 2

Question

Solve each equation or inequality. Round to the nearest ten-thousandth. \(\ln 2 x=4\)

Step-by-Step Solution

Answer

The value of \( x \) is approximately 27.2991.

Step 1: Understand the Equation

We start with the equation \( \ln 2x = 4 \). The natural logarithm \( \ln \) implies that the base of the logarithm is \( e \), where \( e \approx 2.71828 \). Our goal is to solve for \( x \).

Step 2: Exponentiate Both Sides to Remove the Logarithm

To remove the natural logarithm, we exponentiate both sides of the equation. This means we raise \( e \) to the power of each side:\[e^{\ln 2x} = e^4 \].

Step 3: Simplify the Exponentiation

Since the exponential function and the natural logarithm function are inverse functions, they cancel each other out. Thus, we have:\[ 2x = e^4 \].

Step 4: Calculate \( e^4 \)

Now, we calculate \( e^4 \) using a calculator:\[ e^4 \approx 54.5982 \] (rounded to the nearest ten-thousandth).

Step 5: Solve for \( x \)

Divide both sides of the equation by 2 to isolate \( x \):\[x = \frac{54.5982}{2} \approx 27.2991 \].

Key Concepts

Understanding the Natural Logarithm

The natural logarithm, represented by \( \ln \), is a special type of logarithm where the base is the mathematical constant \( e \). The value of \( e \) is approximately 2.71828. It's a unique number that's commonly used in mathematics, especially in calculus and exponential growth problems.
When we see an expression like \( \ln 2x = 4 \), it signifies that the power to which we must raise \( e \) to get \( 2x \) is 4. Solving such equations involve understanding that \( \ln \) and \( e \) are linked through exponentiation and are essentially inverse operations.

• Natural logarithm is used to solve equations involving exponential growth.
• Becomes crucial in solving real-world finance, physics, and engineering problems.

When dealing with a natural logarithm equation, it's key to recognize the relation between the logarithm and its base \( e \). The journey to solution leads us through exponentiation and understanding inverse relationships.

The Process of Exponentiation

Exponentiation is the operation of raising a number to a certain power. When we exponentiate both sides of the equation \( \ln 2x = 4 \), we raise \( e \) to the power on each side, resulting in \( e^{\ln 2x} = e^4 \).
This operation essentially undoes the natural logarithm because the natural logarithm and the exponential function are inverse functions. Thus, utilizing exponentiation is critical to solving logarithmic equations.

• Exponentiation cancels out the natural logarithm in this specific equation.
• Convert the logarithmic form to an exponential form to simplify the equation.

By exponentiating, the natural log disappears, allowing the equation to be expressed in a simpler algebraic form, equipped for straightforward algebraic techniques.

Inverse Functions in Action

Inverse functions are pairs of functions that reverse each other's operations. In our equation, \( \ln \, \text{and} \, e^x \) are inverse functions. The natural logarithm \( \ln \, x \) and its inverse \( e^x \) allow us to solve the equation by undoing each other's operations
When \( e \) is raised to the power of the natural log, the result is the original value before the log was applied. In practical terms, this means \( e^{\ln 2x} \) simplifies directly to \( 2x \) because they undo each other.

• Inverse functions cancel each other out, simplifying expressions.
• Understanding inverse relationships is essential for grasping complex equations like logarithms.

Recognizing that logarithms and exponentials reverse each other's operations is key to solving and simplifying logarithmic equations efficiently.

Importance of Rounding

Rounding is a mathematical technique used to approximate a decimal to its nearest value at a particular decimal place. In the solution, we calculate \( e^4 \), which is approximately 54.5982 when rounded to the nearest ten-thousandth. Rounding is vital because it provides a manageable number to work with, especially when exact values aren't needed or when calculations need to be simplified.

• Rounding helps provide clearer, more concise answers.
• Useful in final answers where exact precision isn't required.

When we reach the answer \( x \approx 27.2991 \), its rounded form allows us to express the solution in a practical and realistic manner. Accurate rounding is crucial for efficiently utilizing numbers in everyday scenarios, like financial and engineering computations.