Solve: E 2 X + 5 = 4 E^{2x+5}=4 E 2 X + 5 = 4 A. X = − 1 2 X=-\frac{1}{2} X = − 2 1 ​ B. X = Ln ⁡ 4 2 − 5 X=\frac{\ln 4}{2}-5 X = 2 L N 4 ​ − 5 C. X = ( Ln ⁡ 4 ) − 5 2 X=\frac{(\ln 4)-5}{2} X = 2 ( L N 4 ) − 5 ​

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Introduction

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Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of exponential functions and their properties. In this article, we will focus on solving the equation $e^{2x+5}=4$, which is a classic example of an exponential equation. We will break down the solution into manageable steps, using a combination of algebraic manipulations and properties of exponential functions.

Step 1: Isolate the Exponential Term

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The first step in solving the equation $e^{2x+5}=4$ is to isolate the exponential term. We can do this by dividing both sides of the equation by $e^{2x+5}$, which gives us:

$$e^{2x+5} \cdot \frac{1}{e^{2x+5}} = 4 \cdot \frac{1}{e^{2x+5}}$$

This simplifies to:

$$1 = 4 \cdot \frac{1}{e^{2x+5}}$$

Step 2: Simplify the Equation

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Now that we have isolated the exponential term, we can simplify the equation further. We can start by dividing both sides of the equation by 4, which gives us:

$$\frac{1}{4} = \frac{1}{e^{2x+5}}$$

Step 3: Take the Natural Logarithm

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The next step is to take the natural logarithm of both sides of the equation. This will allow us to eliminate the exponential term and solve for $x$. We can use the property of logarithms that states $\ln(a^b) = b \cdot \ln(a)$.

$$\ln\left(\frac{1}{4}\right) = \ln\left(\frac{1}{e^{2x+5}}\right)$$

This simplifies to:

$$\ln\left(\frac{1}{4}\right) = -(2x+5)$$

Step 4: Simplify the Equation

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Now that we have taken the natural logarithm of both sides of the equation, we can simplify it further. We can start by evaluating the natural logarithm of $\frac{1}{4}$, which is equal to $-\ln(4)$.

$$-\ln(4) = -(2x+5)$$

Step 5: Solve for x

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The final step is to solve for $x$. We can start by dividing both sides of the equation by $-1$, which gives us:

$$\ln(4) = 2x+5$$

Next, we can subtract 5 from both sides of the equation, which gives us:

$$\ln(4) - 5 = 2x$$

Finally, we can divide both sides of the equation by 2, which gives us:

$$\frac{\ln(4) - 5}{2} = x$$

Conclusion

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In this article, we have solved the equation $e^{2x+5}=4$ 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 term and ending with solving for $x$. The final solution is:

$$x = \frac{\ln(4) - 5}{2}$$

This solution is consistent with the options provided, which are:

A. $x = -\frac{1}{2}$ B. $x = \frac{\ln(4)}{2} - 5$ C. $x = \frac{(\ln(4)) - 5}{2}$

The correct solution is option C, which is:

$$x = \frac{(\ln(4)) - 5}{2}$$

Frequently Asked Questions

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Q: What is the definition of an exponential equation?

A: An exponential equation is an equation that involves an exponential function, which is a function of the form $f(x) = a^x$, where $a$ is a positive constant.

Q: What is the property of logarithms that we used in this article?

A: The property of logarithms that we used in this article is $\ln(a^b) = b \cdot \ln(a)$.

Q: What is the final solution to the equation $e^{2x+5}=4$?

A: The final solution to the equation $e^{2x+5}=4$ is:

$$x = \frac{(\ln(4)) - 5}{2}$$

References

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• [1] "Exponential Functions" by Math Is Fun
• [2] "Logarithmic Functions" by Math Is Fun
• [3] "Solving Exponential Equations" by Khan Academy
  

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Introduction

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In our previous article, we solved the equation $e^{2x+5}=4$ using a combination of algebraic manipulations and properties of exponential functions. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving exponential equations.

Q&A

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Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential function, which is a function of the form $f(x) = a^x$, where $a$ is a positive constant.

Q: What are some common types of exponential equations?

A: Some common types of exponential equations include:

• Equations with a single exponential term, such as $e^{2x+5}=4$
• Equations with multiple exponential terms, such as $e^{2x+5} \cdot e^{3x-2} = 4$
• Equations with exponential terms and other functions, such as $e^{2x+5} + 3x = 4$

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can follow these steps:

1. Isolate the exponential term
2. Take the natural logarithm of both sides of the equation
3. Simplify the equation using properties of logarithms
4. Solve for the variable

Q: What is the property of logarithms that I should use when solving exponential equations?

A: The property of logarithms that you should use when solving exponential equations is $\ln(a^b) = b \cdot \ln(a)$.

Q: How do I use the property of logarithms to simplify an exponential equation?

A: To use the property of logarithms to simplify an exponential equation, you can follow these steps:

1. Take the natural logarithm of both sides of the equation
2. Use the property of logarithms to rewrite the exponential term
3. Simplify the equation using algebraic manipulations

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 of the equation
• Not using the property of logarithms to simplify the equation
• 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 follow these steps:

1. Plug the solution back into the original equation
2. Check if the equation is true
3. If the equation is not true, then the solution is an extraneous solution

Examples

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Example 1: Solving the equation $e^{2x+5}=4$

To solve the equation $e^{2x+5}=4$, we can follow the steps outlined above:

1. Isolate the exponential term: $e^{2x+5} = 4$
2. Take the natural logarithm of both sides of the equation: $\ln(e^{2x+5}) = \ln(4)$
3. Simplify the equation using properties of logarithms: $2x+5 = \ln(4)$
4. Solve for the variable: $x = \frac{\ln(4) - 5}{2}$

Example 2: Solving the equation $e^{2x+5} \cdot e^{3x-2} = 4$

To solve the equation $e^{2x+5} \cdot e^{3x-2} = 4$, we can follow the steps outlined above:

1. Isolate the exponential term: $e^{2x+5} \cdot e^{3x-2} = 4$
2. Take the natural logarithm of both sides of the equation: $\ln(e^{2x+5} \cdot e^{3x-2}) = \ln(4)$
3. Simplify the equation using properties of logarithms: $(2x+5) + (3x-2) = \ln(4)$
4. Solve for the variable: $x = \frac{\ln(4) + 3}{5}$

Conclusion

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In this article, we have provided a Q&A guide to help you better understand the concepts and techniques involved in solving exponential equations. We have covered topics such as the definition of an exponential equation, common types of exponential equations, and how to solve exponential equations using algebraic manipulations and properties of logarithms. We have also provided examples to illustrate the concepts and techniques outlined in this article.

Frequently Asked Questions

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Q: What is the definition of an exponential equation?

A: An exponential equation is an equation that involves an exponential function, which is a function of the form $f(x) = a^x$, where $a$ is a positive constant.

Q: What are some common types of exponential equations?

A: Some common types of exponential equations include:

• Equations with a single exponential term, such as $e^{2x+5}=4$
• Equations with multiple exponential terms, such as $e^{2x+5} \cdot e^{3x-2} = 4$
• Equations with exponential terms and other functions, such as $e^{2x+5} + 3x = 4$

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can follow these steps:

1. Isolate the exponential term
2. Take the natural logarithm of both sides of the equation
3. Simplify the equation using properties of logarithms
4. Solve for the variable

References

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• [1] "Exponential Functions" by Math Is Fun
• [2] "Logarithmic Functions" by Math Is Fun
• [3] "Solving Exponential Equations" by Khan Academy