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

Introduction

Exponential equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving the equation $8(e)^{2x+1} = 4$ and provide a step-by-step guide on how to arrive at the correct solution.

Understanding Exponential Equations

Exponential equations involve variables in the exponent, and they can be solved using logarithms. The general form of an exponential equation is $a^x = b$, where $a$ is the base, $x$ is the variable, and $b$ is the constant. In our equation, $8(e)^{2x+1} = 4$, the base is $e$, the variable is $x$, and the constant is $4$.

Step 1: Isolate the Exponential Term

To solve the equation, we need to isolate the exponential term. We can do this by dividing both sides of the equation by $8$:

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

$$\frac{8(e)^{2x+1}}{8} = \frac{4}{8}$$

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

Step 2: Take the Natural Logarithm

Now that we have isolated the exponential term, we can take the natural logarithm of both sides of the equation. The natural logarithm is denoted by $\ln$, and it is the inverse of the exponential function. Taking the natural logarithm of both sides 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

Now that we have simplified the equation, we can solve for $x$. We can do this by isolating $x$ on one side of the equation:

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

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

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

Conclusion

In this article, we have solved the equation $8(e)^{2x+1} = 4$ using the natural logarithm. We have taken the natural logarithm of both sides of the equation, isolated $x$ on one side, and arrived at the solution $x = \frac{\ln\left(\frac{1}{2}\right) - 1}{2}$. This solution is the correct answer to the equation.

Discussion

The solution to the equation $8(e)^{2x+1} = 4$ is $x = \frac{\ln\left(\frac{1}{2}\right) - 1}{2}$. This solution can be verified by plugging it back into the original equation. If you have any questions or need further clarification, please feel free to ask.

Answer

The correct answer to the equation $8(e)^{2x+1} = 4$ is:

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

Final Answer

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Q: What is an exponential equation?

A: An exponential equation is an equation that involves a variable in the exponent. It is a mathematical statement that contains a variable raised to a power, and it can be solved using logarithms.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the exponential term, take the natural logarithm of both sides, and then solve for the variable. This process involves using the properties of logarithms and the inverse relationship between the exponential function and the natural logarithm.

Q: What is the natural logarithm?

A: The natural logarithm is a mathematical function that is the inverse of the exponential function. It is denoted by $\ln$ and is used to solve exponential equations. The natural logarithm of a number $x$ is the power to which the base $e$ must be raised to produce $x$.

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

A: To use the natural logarithm to solve an exponential equation, you need to take the natural logarithm of both sides of the equation. This will allow you to isolate the variable and solve for it. The natural logarithm can be used to solve exponential equations with any base, but it is most commonly used with the base $e$.

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

A: The natural logarithm and the common logarithm are two different mathematical functions that are used to solve exponential equations. The natural logarithm is denoted by $\ln$ and is the inverse of the exponential function with base $e$. The common logarithm is denoted by $\log$ and is the inverse of the exponential function with base $10$. The natural logarithm is used more commonly in mathematics and science, while the common logarithm is used more commonly in engineering and finance.

Q: Can I use the common logarithm to solve an exponential equation?

A: Yes, you can use the common logarithm to solve an exponential equation, but it is not as commonly used as the natural logarithm. The common logarithm is used to solve exponential equations with base $10$, and it can be used to solve equations with other bases as well.

Q: How do I choose between the natural logarithm and the common logarithm?

A: When choosing between the natural logarithm and the common logarithm, you need to consider the base of the exponential equation. If the base is $e$, you should use the natural logarithm. If the base is $10$, you should use the common logarithm. If the base is not $e$ or $10$, you can use either the natural logarithm or the common logarithm, depending on which one is more convenient.

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

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

• Not isolating the exponential term
• Not taking the natural logarithm of both sides
• Not using the correct base for the logarithm
• Not simplifying the equation correctly
• Not checking the solution to make sure it is correct

Q: How do I check my solution to an exponential equation?

A: To check your solution to an exponential equation, you need to plug it back into the original equation and make sure it is true. This will help you to verify that your solution is correct and to make sure that you have not made any mistakes.

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

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

• Population growth and decline
• Compound interest and finance
• Radioactive decay and nuclear physics
• Chemical reactions and kinetics
• Electrical circuits and electronics

Q: How do I use exponential equations in real-world applications?

A: To use exponential equations in real-world applications, you need to understand the underlying principles and concepts. You need to be able to set up and solve exponential equations, and you need to be able to interpret the results in the context of the real-world problem. This requires a strong understanding of mathematics and science, as well as the ability to apply mathematical concepts to real-world problems.

Conclusion

Exponential equations are a fundamental concept in mathematics and science, and they have many real-world applications. To solve exponential equations, you need to understand the properties of logarithms and the inverse relationship between the exponential function and the natural logarithm. By following the steps outlined in this article, you can solve exponential equations and apply them to real-world problems.