Problem 92 - College Algebra with Modeling and Visualization

Question

Solve each equation. Approximate answers to four decimal places when appropriate. $$2 \ln 3 x=8$$

Step-by-Step Solution

Answer

\(x \approx 18.1994\)

Step 1: Isolate the logarithmic term

First, divide both sides of the equation by 2 to isolate the logarithmic term \(\ln(3x)\). This gives us: \[ \ln(3x) = 4 \]

Step 2: Exponentiate to remove the natural log

Exponentiate both sides of the equation to eliminate the natural logarithm, using the property that if \(\ln(y) = a\), then \(y = e^a\). This results in: \[ 3x = e^4 \]

Step 3: Solve for x

Now divide both sides by 3 to solve for \(x\): \[ x = \frac{e^4}{3} \]

Step 4: Calculate the numerical value

Calculate \(e^4\) using a calculator and then divide by 3. \(e^4 \approx 54.5982\), so: \[ x = \frac{54.5982}{3} \approx 18.1994 \]

Key Concepts

Natural logarithm

Natural logarithms are logarithms with base \(e\), where \(e\) is a special number approximately equal to 2.71828. In a natural logarithm, the notation \(\ln(x)\) represents the logarithmic operation. The natural log tells us what power we need to raise \(e\) to obtain a certain number.

For example, if \(\ln(y) = a\), it means that \(e^a = y\). Natural logarithms are especially useful in calculus and exponential growth problems, where the base \(e\) often appears naturally.

When solving logarithmic equations, it's important to isolate the logarithmic term. In our example equation, we first isolate \(\ln(3x)\) by dividing both sides by 2. This is a common technique to simplify equations and prepare them for further solving steps.

Exponentiation

Exponentiation is a mathematical operation where a number, known as the base, is raised to a certain power. For natural logs, we use \(e\) as the base. By exponentiating both sides of a logarithmic equation, we can "undo" the logarithm and solve for the variable. The property of exponents allows us to transform an equation like \(\ln(y) = a\) into \(y = e^a\).

In our exercise, after isolating the logarithmic term \(\ln(3x) = 4\), we exponentiate both sides to solve for \(3x\). This step lets us convert the logarithmic equation into the more straightforward form of \(3x = e^4\). This transformation is crucial as it moves us from a logarithmic problem to an exponential problem, simplifying the process of finding the unknown variable.

Numerical approximation

Numerical approximation involves estimating a numerical solution to a problem with a specific degree of accuracy. Many mathematical values, like \(e^4\) in our exercise, do not simplify neatly and must be approximated using a calculator or computational tool.

For the equation \(3x = e^4\), calculating the actual value of \(e^4\) gives us approximately 54.5982. We then divide this result by 3 to solve for \(x\), giving \(x \approx 18.1994\). This step illustrates the practical importance of numerical approximation in obtaining precise answers, especially when exact solutions involve irrational numbers.

• Ensure your calculator is in the correct mode for finding exponentials.
• Round the result as necessary to match the required precision.

Numerical approximations help us work with complex numbers in a manageable way, especially in real-world problems where exact values might not be available.