Problem 44 - Algebra A Combined Function

Question

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ \ln (2 x+5)=3.4 $$

Step-by-Step Solution

Answer

Exact: \( x = \frac{e^{3.4} - 5}{2} \); Approximation: \( x \approx 12.4821 \).

Step 1: Isolate the natural logarithm

The equation is \( \ln(2x + 5) = 3.4 \). In this problem, the natural logarithm, \( \ln(2x + 5) \), is already isolated on one side of the equation.

Step 2: Exponentiate to eliminate the logarithm

To eliminate the natural logarithm, we will exponentiate both sides using the base of the natural logarithm, which is \( e \). Thus, we have \( e^{\ln(2x + 5)} = e^{3.4} \). The left side simplifies to \( 2x + 5 \), resulting in \( 2x + 5 = e^{3.4} \).

Step 3: Calculate \( e^{3.4} \)

Use a calculator to find \( e^{3.4} \). The approximate value is \( e^{3.4} \approx 29.9641 \). Now substitute this back into the equation to get \( 2x + 5 = 29.9641 \).

Step 4: Solve for \( x \)

Subtract 5 from both sides to isolate the term with \( x \): \( 2x = 29.9641 - 5 \), which simplifies to \( 2x = 24.9641 \).

Step 5: Final calculation for \( x \)

Divide both sides by 2 to solve for \( x \): \( x = \frac{24.9641}{2} \). Thus, \( x = 12.48205 \). For practical purposes, we round this to four decimal places to get \( x \approx 12.4821 \).

Key Concepts

Understanding the Natural Logarithm

The natural logarithm is a mathematical function denoted as \( \ln \). It is the inverse of the exponential function with the base \( e \), where \( e \approx 2.71828 \). In simpler terms, if you have an equation involving \( \ln \), and you exponentiate it using \( e \), you will convert it back into its original form. The natural logarithm is widely used in various fields such as science, engineering, and finance, providing solutions to problems involving growth processes.
In the original exercise, the expression \( \ln(2x + 5) = 3.4 \) implies that we are dealing with the logarithmic inverse of some exponential function. To solve such equations, our primary goal is first to isolate the logarithm, which is already done in this equation. Knowing how to efficiently handle natural logarithms is key to solving complex logarithmic equations.

Exponentiation Explained

Exponentiation is the process of raising a number to a power. In this context, we use it to counteract the effect of the natural logarithm by applying the base \( e \) to both sides of the equation. This is because the natural logarithm \( \ln \) and exponentiation with base \( e \) are inverse operations.
In our exercise, we started with \( \ln(2x + 5) = 3.4 \) and applied exponentiation to transform it to \( e^{\ln(2x + 5)} = e^{3.4} \). The left side simplifies as \( 2x + 5 \), as \( e^{\ln(y)} = y \) for any expression \( y \). Understanding exponentiation in this context allows us to convert logarithmic equations back into a form that can be easily solved.

Providing an Exact Solution

An exact solution refers to the process of solving an equation using precise analytical methods without rounding until the final step. It remains in its true mathematical form.
For our problem, after exponentiating, we expressed \( 2x + 5 = e^{3.4} \). To find \( x \), we rearrange to get \( 2x = e^{3.4} - 5 \), and then solve for \( x \) by dividing the entire expression by 2. This gives us \( x = \frac{e^{3.4} - 5}{2} \), which is the exact solution. No approximations were made here, which is crucial for maintaining mathematical accuracy.

Making Approximations

Approximations come in handy when we need a solution that is practical for everyday use. It involves rounding numbers to a certain number of decimal places, making calculations easier to handle without computing extensive digits.
In this exercise, after calculating \( e^{3.4} \), we found that its approximate value was \( 29.9641 \). Using this approximation, the equation \( 2x + 5 = 29.9641 \) was solved for \( x \), leading to \( x \approx 12.48205 \). We then rounded this to four decimal places, yielding \( x \approx 12.4821 \). Approximation is significant in real-world scenarios where exact numbers are cumbersome and unnecessary.