Select The Correct Answer.What Is The Solution To This Equation?${ 8(e)^{2x+1} = 4 }$A. ${ X = \frac{\ln (0.5)}{2} - 1 }$B. ${ X = \frac{\ln (0.5) - 1}{2} }$C. ${ X = \frac{\ln (0.5)}{2} + 1 }$D. $[ X =

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic manipulations and logarithmic properties. In this article, we will focus on solving the equation $8(e)^{2x+1} = 4$ and provide a step-by-step guide to help you arrive at the correct solution.

Understanding Exponential Equations

Exponential equations involve variables in the exponent, and they can be challenging to solve. However, with the right approach and techniques, you can master the art of solving exponential equations. In this section, we will discuss the key concepts and properties that you need to know to solve exponential equations.

The Equation $8(e)^{2x+1} = 4$

The given equation is $8(e)^{2x+1} = 4$. To solve this equation, we need to isolate the variable $x$. We can start by simplifying the equation using the properties of exponents.

Step 1: Simplify the Equation

We can simplify the equation by dividing both sides by $8$. This gives us:

$$e^{2x+1} = \frac{1}{2}$$

Step 2: Take the Natural Logarithm

To solve for $x$, we can take the natural logarithm (ln) of both sides of the equation. This gives us:

$$\ln(e^{2x+1}) = \ln\left(\frac{1}{2}\right)$$

Using the property of logarithms that states $\ln(a^b) = b\ln(a)$, we can simplify the left-hand side of the equation:

$$(2x+1)\ln(e) = \ln\left(\frac{1}{2}\right)$$

Since $\ln(e) = 1$, we can simplify the equation further:

$$2x+1 = \ln\left(\frac{1}{2}\right)$$

Step 3: Solve for $x$

To solve for $x$, we can isolate the variable by subtracting $1$ from both sides of the equation:

$$2x = \ln\left(\frac{1}{2}\right) - 1$$

Finally, we can divide both sides by $2$ to solve for $x$:

$$x = \frac{\ln\left(\frac{1}{2}\right) - 1}{2}$$

Conclusion

In this article, we solved the equation $8(e)^{2x+1} = 4$ using a step-by-step approach. We simplified the equation, took the natural logarithm of both sides, and finally solved for $x$. The correct solution is:

$$x = \frac{\ln\left(\frac{1}{2}\right) - 1}{2}$$

Answer Options

Here are the answer options:

A. $x = \frac{\ln(0.5)}{2} - 1$ B. $x = \frac{\ln(0.5) - 1}{2}$ C. $x = \frac{\ln(0.5)}{2} + 1$ D. $x = \frac{\ln(0.5) + 1}{2}$

Which answer is correct?

The correct answer is B. $x = \frac{\ln(0.5) - 1}{2}$.

Why is this answer correct?

This answer is correct because we solved the equation using a step-by-step approach, and the final solution matches the answer option B. The other answer options are incorrect because they do not match the solution we obtained.

Final Thoughts

Solving exponential equations requires a deep understanding of algebraic manipulations and logarithmic properties. In this article, we provided a step-by-step guide to solving the equation $8(e)^{2x+1} = 4$. We simplified the equation, took the natural logarithm of both sides, and finally solved for $x$. The correct solution is:

$$x = \frac{\ln\left(\frac{1}{2}\right) - 1}{2}$$

Q: What is an exponential equation?

A: An exponential equation is an equation that involves a variable in the exponent. It is a type of equation that can be challenging to solve, but with the right approach and techniques, you can master the art of solving exponential equations.

Q: What are some common properties of exponential equations?

A: Some common properties of exponential equations include:

• The product rule: $a^m \cdot a^n = a^{m+n}$
• The quotient rule: $\frac{a^m}{a^n} = a^{m-n}$
• The power rule: $(a^m)^n = a^{mn}$
• The property of logarithms: $\ln(a^b) = b\ln(a)$

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, you can use the properties of exponents to combine like terms and isolate the variable. For example, if you have the equation $2^x \cdot 2^y = 2^{x+y}$, you can simplify it by combining the like terms.

Q: What is the natural logarithm (ln)?

A: The natural logarithm (ln) is a logarithmic function that is the inverse of the exponential function. It is denoted by the symbol $\ln$ and is defined as the logarithm of a number to the base $e$. The natural logarithm is used to solve exponential equations and is a fundamental concept in mathematics.

Q: How do I solve an exponential equation using the natural logarithm?

A: To solve an exponential equation using the natural logarithm, you can take the natural logarithm of both sides of the equation and then use the property of logarithms to simplify the equation. For example, if you have the equation $e^x = 2$, you can take the natural logarithm of both sides and then use the property of logarithms to solve for $x$.

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

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

• Not using the correct properties of exponents
• Not isolating the variable correctly
• Not using the natural logarithm correctly
• Not checking the solution for extraneous solutions

Q: How do I check my solution for extraneous solutions?

A: To check your solution for extraneous solutions, you can plug the solution back into the original equation and check if it is true. If the solution is not true, then it is an extraneous solution and you should discard it.

Q: What are some real-world applications of exponential equations?

A: Exponential equations have many real-world applications, including:

• Modeling population growth
• Modeling chemical reactions
• Modeling financial investments
• Modeling electrical circuits

Q: How can I practice solving exponential equations?

A: You can practice solving exponential equations by working through practice problems and exercises. You can also use online resources and tools to help you practice and improve your skills.

Conclusion

Solving exponential equations requires a deep understanding of algebraic manipulations and logarithmic properties. By following the steps outlined in this article and practicing regularly, you can master the art of solving exponential equations and apply it to real-world problems. Remember to always check your solution for extraneous solutions and to use the correct properties of exponents and logarithms.