Solve For \[$ X \$\].$\[ E^{2x} = E^{3x-1} \\]

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific type of exponential equation, namely $e^{2x} = e^{3x-1}$. We will break down the solution into manageable steps, using a combination of algebraic manipulations and properties of exponential functions.

Understanding Exponential Functions

Before we dive into the solution, let's take a moment to understand the properties of exponential functions. The exponential function $f(x) = e^x$ is a continuous and differentiable function that has several important properties. One of the key properties is that the exponential function is one-to-one, meaning that if $f(x) = f(y)$, then $x = y$. This property will be crucial in solving the given equation.

Step 1: Isolate the Exponential Terms

The first step in solving the equation $e^{2x} = e^{3x-1}$ is to isolate the exponential terms on one side of the equation. We can do this by subtracting $e^{3x-1}$ from both sides of the equation, resulting in:

$$e^{2x} - e^{3x-1} = 0$$

Step 2: Factor Out the Common Term

The next step is to factor out the common term $e^{3x-1}$ from the left-hand side of the equation. We can do this by recognizing that $e^{2x} = e^{3x-1} \cdot e^{-(3x-1)}$. Therefore, we can rewrite the equation as:

$$e^{3x-1}(e^{-(3x-1)} - 1) = 0$$

Step 3: Set Each Factor Equal to Zero

Now that we have factored the equation, we can set each factor equal to zero and solve for $x$. The first factor, $e^{3x-1}$, is always positive, so it cannot be equal to zero. Therefore, we can focus on the second factor, $e^{-(3x-1)} - 1$. Setting this factor equal to zero, we get:

$$e^{-(3x-1)} - 1 = 0$$

Step 4: Solve for x

To solve for $x$, we can start by adding 1 to both sides of the equation, resulting in:

$$e^{-(3x-1)} = 1$$

Next, we can take the natural logarithm of both sides of the equation, resulting in:

$$-(3x-1) = \ln(1)$$

Since $\ln(1) = 0$, we can simplify the equation to:

$$-(3x-1) = 0$$

Step 5: Solve for x (continued)

To solve for $x$, we can start by adding $3x$ to both sides of the equation, resulting in:

$$-1 = 3x$$

Next, we can divide both sides of the equation by 3, resulting in:

$$x = -\frac{1}{3}$$

Conclusion

In this article, we have solved the exponential equation $e^{2x} = e^{3x-1}$ using a combination of algebraic manipulations and properties of exponential functions. We have broken down the solution into manageable steps, starting with isolating the exponential terms and ending with solving for $x$. The final solution is $x = -\frac{1}{3}$.

Additional Tips and Tricks

• When solving exponential equations, it's essential to isolate the exponential terms on one side of the equation.
• Use properties of exponential functions, such as the one-to-one property, to simplify the equation.
• Factor out common terms to make the equation more manageable.
• Set each factor equal to zero and solve for $x$.

Common Mistakes to Avoid

• Don't forget to isolate the exponential terms on one side of the equation.
• Avoid using the wrong properties of exponential functions, such as the property that $e^x = e^y$ implies $x = y$.
• Don't forget to factor out common terms to make the equation more manageable.

Real-World Applications

Exponential equations have numerous real-world applications, including:

• Modeling population growth and decay
• Analyzing chemical reactions
• Solving problems in finance and economics

By mastering the skills of solving exponential equations, you can apply them to a wide range of real-world problems and make informed decisions in various fields.

Final Thoughts

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Introduction

In our previous article, we explored the steps to solve exponential equations, including the equation $e^{2x} = e^{3x-1}$. However, we know that practice makes perfect, and there's no better way to reinforce your understanding than by answering questions and solving more problems. In this article, we'll provide a Q&A guide to help you solidify your skills in solving exponential equations.

Q1: What is the first step in solving an exponential equation?

A1: The first step in solving an exponential equation is to isolate the exponential terms on one side of the equation. This can be done by subtracting or adding the same value to both sides of the equation.

Q2: How do I know which property of exponential functions to use?

A2: When solving exponential equations, you can use the following properties of exponential functions:

• The one-to-one property: If $f(x) = f(y)$, then $x = y$.
• The property of equality: If $f(x) = f(y)$, then $x = y$.
• The property of multiplication: If $f(x) = f(y)$, then $f(x) \cdot f(z) = f(y) \cdot f(z)$.

Q3: What is the difference between $e^x$ and $e^{-x}$?

A3: $e^x$ and $e^{-x}$ are two different exponential functions. $e^x$ represents the exponential function with a base of $e$ and an exponent of $x$, while $e^{-x}$ represents the exponential function with a base of $e$ and an exponent of $-x$.

Q4: How do I factor out common terms in an exponential equation?

A4: To factor out common terms in an exponential equation, you can use the distributive property of multiplication over addition. For example, if you have the equation $e^{2x} + e^{3x} = e^{4x}$, you can factor out the common term $e^{2x}$ by rewriting the equation as $e^{2x}(1 + e^{x}) = e^{4x}$.

Q5: What is the final step in solving an exponential equation?

A5: The final step in solving an exponential equation is to set each factor equal to zero and solve for $x$. This can be done by using the properties of exponential functions, such as the one-to-one property, to simplify the equation.

Q6: Can I use logarithms to solve exponential equations?

A6: Yes, you can use logarithms to solve exponential equations. By taking the logarithm of both sides of the equation, you can simplify the equation and solve for $x$. For example, if you have the equation $e^{2x} = e^{3x-1}$, you can take the natural logarithm of both sides of the equation to get $2x = 3x - 1$.

Q7: How do I know if an exponential equation has a solution?

A7: To determine if an exponential equation has a solution, you can use the following criteria:

• The equation must have a real solution.
• The equation must have a unique solution.
• The equation must have a solution that satisfies the given conditions.

Q8: Can I use exponential equations to model real-world problems?

A8: Yes, you can use exponential equations to model real-world problems. Exponential equations can be used to model population growth and decay, chemical reactions, and other phenomena that involve exponential change.

Q9: How do I apply exponential equations to real-world problems?

A9: To apply exponential equations to real-world problems, you can use the following steps:

• Identify the problem and the variables involved.
• Write an exponential equation that models the problem.
• Solve the equation using the steps outlined in this article.
• Interpret the results and apply them to the real-world problem.

Q10: What are some common mistakes to avoid when solving exponential equations?

A10: Some common mistakes to avoid when solving exponential equations include:

• Forgetting to isolate the exponential terms on one side of the equation.
• Using the wrong properties of exponential functions.
• Not factoring out common terms.
• Not setting each factor equal to zero and solving for $x$.

By following the steps outlined in this article and avoiding common mistakes, you can become proficient in solving exponential equations and apply them to real-world problems.