Solving for the Unknown Value of x in a Natural Logarithm Equation

Oct 13, 2025 by recoveryshake.com 0 views

Solving for the Unknown Value of x in a Natural Logarithm Equation

In this article, we will delve into solving a natural logarithm equation for the unknown value of x. The equation given is $$5 \ln (3x - 2) = 15$$. Our goal is to isolate x and find its value, rounding the solution to two decimal places. We will explore the steps involved in solving this equation and provide the correct solution.

Step 1: Isolate the Natural Logarithm

The first step in solving this equation is to isolate the natural logarithm term. We can do this by dividing both sides of the equation by 5.

\frac{5 \ln (3x - 2)}{5} = \frac{15}{5}

This simplifies to:

\ln (3x - 2) = 3

Step 2: Exponentiate Both Sides

To get rid of the natural logarithm, we can exponentiate both sides of the equation. Since the natural logarithm is the inverse of the exponential function, we can use the exponential function to "undo" the natural logarithm.

e^{\ln (3x - 2)} = e^3

This simplifies to:

3x - 2 = e^3

Step 3: Solve for x

Now that we have isolated the term with x, we can solve for x. We can start by adding 2 to both sides of the equation.

3x = e^3 + 2

Next, we can divide both sides of the equation by 3.

x = \frac{e^3 + 2}{3}

Step 4: Calculate the Value of x

To find the value of x, we can calculate the value of the expression on the right-hand side of the equation.

x = \frac{e^3 + 2}{3}

Using a calculator, we can find that:

e^3 \approx 20.0855

Substituting this value into the equation, we get:

x \approx \frac{20.0855 + 2}{3}

Simplifying, we get:

x \approx \frac{22.0855}{3}

x \approx 7.36183

Rounding this value to two decimal places, we get:

x \approx 7.36

In this article, we solved a natural logarithm equation for the unknown value of x. We started by isolating the natural logarithm term, then exponentiated both sides to get rid of the natural logarithm. Finally, we solved for x and calculated its value. The correct solution is x ≈ 7.36.

Let's compare our solution to the options provided:

A. x = 6.03
B. x = 7.36
C. x = 1.24
D. x = 217,936.16

Our solution, x ≈ 7.36, matches option B. Therefore, the correct answer is B. x = 7.36.

The final answer is B. x = 7.36.
Solving for the Unknown Value of x in a Natural Logarithm Equation: Q&A

In our previous article, we solved a natural logarithm equation for the unknown value of x. We received several questions from readers regarding the solution and the steps involved in solving the equation. In this article, we will address some of the most frequently asked questions and provide additional clarification on the solution.

Q: What is the natural logarithm, and how is it used in the equation?

A: The natural logarithm, denoted by ln(x), is the inverse of the exponential function. It is used to solve equations that involve exponential functions. In the equation $$5 \ln (3x - 2) = 15$$, the natural logarithm is used to isolate the term with x.

Q: Why do we need to isolate the natural logarithm term?

A: We need to isolate the natural logarithm term because it is the only term that contains x. By isolating the natural logarithm term, we can use the properties of logarithms to solve for x.

Q: What is the difference between the natural logarithm and the common logarithm?

A: The natural logarithm and the common logarithm are both logarithmic functions, but they have different bases. The natural logarithm has a base of e, while the common logarithm has a base of 10. In this equation, we use the natural logarithm because it is the inverse of the exponential function.

Q: Can we use the common logarithm instead of the natural logarithm?

A: No, we cannot use the common logarithm instead of the natural logarithm in this equation. The common logarithm is not the inverse of the exponential function, and it would not be used to solve this type of equation.

Q: Why do we need to exponentiate both sides of the equation?

A: We need to exponentiate both sides of the equation because it allows us to get rid of the natural logarithm. By exponentiating both sides, we can use the properties of exponents to solve for x.

Q: Can we use a different method to solve the equation?

A: Yes, we can use a different method to solve the equation. However, the method we used in this article is the most straightforward and efficient way to solve the equation.

Q: What is the significance of the value e^3 in the equation?

A: The value e^3 is an important constant in mathematics. It is used to represent the base of the natural logarithm. In this equation, e^3 is used to solve for x.

Q: Can we use a calculator to solve the equation?

A: Yes, we can use a calculator to solve the equation. However, it is also important to understand the steps involved in solving the equation and to be able to explain the solution.

In this article, we addressed some of the most frequently asked questions regarding the solution to the natural logarithm equation. We provided additional clarification on the steps involved in solving the equation and discussed the significance of the value e^3. We hope that this article has been helpful in understanding the solution to the equation.

For additional resources on solving natural logarithm equations, we recommend the following:

Khan Academy: Natural Logarithm
Mathway: Natural Logarithm
Wolfram Alpha: Natural Logarithm

The final answer is B. x = 7.36.