Problem 47 - College Algebra

Question

Exercises \(45-52\) involve equations with natural logarithms. Solve each equation by isolating the natural logarithm and exponentiating both sides. Express the answer in terms of \(e\) Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$5 \ln (2 x)=20$$

Step-by-Step Solution

Answer

The solution to the equation \(5 \ln (2x)=20\) is \(x = e^4/2 \) or approximately \(x = 27.30\).

Step 1: Isolate the natural logarithm

Divide both sides of the equation by 5 to isolate \( \ln (2x)\) on one side. This results in the equation: \( \ln (2x) = 20/5 = 4\).

Step 2: Remove the natural logarithm

To remove the natural logarithm and solve for \(x\), exponentiate both sides using the base \(e\), this gives \( e^{\ln(2x)} = e^4\). The left-hand side simplifies to \(2x\) because \(e\) and \(\ln\) are inverse functions.

Step 3: Solve for \(x\)

Solving for \(x\) requires dividing both sides of the equation by 2. This results in, \(x = e^4/2 \), which is the final answer in terms of \(e\).

Step 4: Decimal Approximation

Using a calculator, \(e^4\) equals about 54.5982. Dividing this result by 2 gives approximately 27.2991. Rounding to two decimal places gives \(x = 27.30\).

Key Concepts

Isolation of Logarithmic Expressions

When solving equations that involve natural logarithms, the first step usually involves isolating the logarithmic expression itself. In this specific example, you start with the equation: \[ 5 \ln(2x) = 20 \]To isolate the natural logarithm, you take the equation and perform the necessary mathematical operations to have the logarithmic term by itself on one side. Here, dividing both sides by 5 achieves this:\[ \ln(2x) = 4 \]This process is important because it sets up the equation in a way that allows you to effectively "undo" the logarithm in the next steps. Isolating the logarithmic expression simplifies the equation and makes it easier to apply further mathematical operations.

Exponentiation to Solve Equations

Exponentiation is a powerful tool used to solve equations involving logarithms. After isolating the logarithmic expression, the goal is to "undo" the logarithm. To do this, you exponentiate both sides of the equation using the base of the logarithm. For natural logarithms, this base is the number \(e\), which is approximately 2.71828. Here's how it looks:After isolating \( \ln(2x) = 4 \), exponentiate to remove the logarithm:\[ e^{\ln(2x)} = e^4 \]Due to the properties of logarithms and exponentials (since \(e\) and \(\ln\) are inverse functions), this simplifies nicely to:\[ 2x = e^4 \]With the natural logarithm effectively "cancelled" out, the equation now becomes much simpler, allowing for straightforward algebraic manipulation to find \(x\). This demonstrates how exponentiation is used to solve for unknowns in equations involving logarithms.

Decimal Approximation Using a Calculator

Mathematics often leads us to neat algebraic solutions, but sometimes a decimal approximation is needed for practical purposes, especially when dealing with irrational numbers like \(e\). In our problem, once we've calculated \(x\) in terms of \(e\), we might need a more tangible numeric answer.After finding \(x = \frac{e^4}{2}\), it’s time to bring in a calculator:- Compute \(e^4\), which is approximately 54.5982- Divide by 2 to find \(x\)This gives us \(x \approx 27.2991\). If required, round to two decimal places:\[ x = 27.30 \]Using a calculator is crucial for finding decimal approximations when the answer involves exponents or irrational numbers. This process bridges the gap between algebraic solutions and utilizable real-world numbers.

Inverse Functions

Inverse functions help us "reverse" mathematical operations, which is notably useful when dealing with equations involving logarithms. In the context of natural logarithms, the inverse function is exponentiation using base \(e\).Consider the equation:\[ \ln(2x) = 4 \]The inverse operation of taking the natural logarithm \(\ln\) is to exponentiate with base \(e\):\[ e^{\ln(2x)} = e^4 \]Through this process, the natural log and the exponential functions cancel each other out (as they are inverses), simplifying the expression to:\[ 2x = e^4 \]Understanding that \(\ln\) and the exponential with base \(e\) are inverse functions is key in solving these logarithmic equations efficiently. It reveals deeper mathematical relationships and allows manipulation of equations to obtain clear, manageable results.