Which Of The Following Is The Solution Of $5 E^{2 X}-4=11$?A. X = Ln ⁡ 3 X=\ln 3 X = Ln 3 B. X = Ln ⁡ 27 X=\ln 27 X = Ln 27 C. X = Ln ⁡ 3 2 X=\frac{\ln 3}{2} X = 2 L N 3 ​ D. X = 3 Ln ⁡ 3 X=\frac{3}{\ln 3} X = L N 3 3 ​

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 exponential equation, $5 e^{2 x}-4=11$, and explore the different solution options provided.

Understanding Exponential Equations

Exponential equations involve the variable in the exponent, and the base is typically a constant. In this case, the base is the natural exponential function, denoted by $e$. The equation $5 e^{2 x}-4=11$ can be rewritten as $5 e^{2 x}=15$, by adding 4 to both sides.

Isolating 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 5, resulting in $e^{2 x}=3$.

Using Properties of Exponents

Now that we have isolated the exponential term, we can use the properties of exponents to simplify the equation. Specifically, we can use the property that $e^{ab}=(ea)^b$.

Applying the Property

Applying the property, we can rewrite the equation as $(e^x)2=3$. Taking the square root of both sides, we get $e^x=\pm\sqrt{3}$.

Solving for x

Now that we have isolated the exponential term, we can solve for x. We can take the natural logarithm (ln) of both sides to get $x=\ln(\pm\sqrt{3})$.

Evaluating the Solutions

We are given four solution options: $x=\ln 3$, $x=\ln 27$, $x=\frac{\ln 3}{2}$, and $x=\frac{3}{\ln 3}$. Let's evaluate each option to determine which one is the correct solution.

Option A: $x=\ln 3$

Option A is $x=\ln 3$. We can evaluate this option by plugging it into the original equation. Substituting $x=\ln 3$ into the equation $5 e^{2 x}-4=11$, we get $5 e^{2 \ln 3}-4=11$. Simplifying the equation, we get $5 (3)^2-4=11$, which is true.

Option B: $x=\ln 27$

Option B is $x=\ln 27$. We can evaluate this option by plugging it into the original equation. Substituting $x=\ln 27$ into the equation $5 e^{2 x}-4=11$, we get $5 e^{2 \ln 27}-4=11$. Simplifying the equation, we get $5 (27)^2-4=11$, which is not true.

Option C: $x=\frac{\ln 3}{2}$

Option C is $x=\frac{\ln 3}{2}$. We can evaluate this option by plugging it into the original equation. Substituting $x=\frac{\ln 3}{2}$ into the equation $5 e^{2 x}-4=11$, we get $5 e^{2 \frac{\ln 3}{2}}-4=11$. Simplifying the equation, we get $5 (3)^1-4=11$, which is not true.

Option D: $x=\frac{3}{\ln 3}$

Option D is $x=\frac{3}{\ln 3}$. We can evaluate this option by plugging it into the original equation. Substituting $x=\frac{3}{\ln 3}$ into the equation $5 e^{2 x}-4=11$, we get $5 e^{2 \frac{3}{\ln 3}}-4=11$. Simplifying the equation, we get $5 (3)^{\frac{1}{\ln 3}}-4=11$, which is not true.

Conclusion

Based on our evaluation of the solution options, we can conclude that the correct solution is $x=\ln 3$. This option satisfies the original equation, and the other options do not.

Final Answer

$$

Introduction

In our previous article, we explored the solution to the exponential equation $5 e^{2 x}-4=11$. We evaluated four different solution options and determined that the correct solution is $x=\ln 3$. 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: What is an exponential equation?

A: An exponential equation is an equation that involves the variable in the exponent. The base is typically a constant, and the variable is raised to a power.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the exponential term. You can do this by using properties of exponents, such as the property that $e^{ab}=(ea)^b$. Once you have isolated the exponential term, you can use logarithms to solve for the variable.

Q: What is the difference between a logarithmic and exponential equation?

A: A logarithmic equation is an equation that involves the logarithm of a variable, while an exponential equation is an equation that involves the variable in the exponent. For example, the equation $\log_2 x=3$ is a logarithmic equation, while the equation $2^x=8$ is an exponential equation.

Q: How do I evaluate the solutions to an exponential equation?

A: To evaluate the solutions to an exponential equation, you need to plug each solution option into the original equation and determine whether it is true or false. You can use algebraic manipulations and properties of exponents to simplify the equation and determine whether it is true or false.

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 using properties of exponents correctly
• Not using logarithms to solve for the variable
• Not evaluating the solutions correctly

Q: How do I apply the properties of exponents to solve exponential equations?

A: To apply the properties of exponents to solve exponential equations, you need to use the following properties:

• $$e^{ab}=(e^a)^b$$

• $$e^{a+b}=e^a \cdot e^b$$

• $$e^{a-b}=\frac{e^a}{e^b}$$

You can use these properties to simplify the equation and isolate the exponential term.

Q: What is the natural logarithm, and how do I use it to solve exponential equations?

A: The natural logarithm is the logarithm of a number to the base e. It is denoted by ln. You can use the natural logarithm to solve exponential equations by taking the natural logarithm of both sides of the equation. This will allow you to isolate the variable and solve for it.

Q: How do I use a calculator to solve exponential equations?

A: To use a calculator to solve exponential equations, you need to enter the equation into the calculator and use the exponent key to raise the base to the power of the variable. You can also use the logarithm key to take the natural logarithm of both sides of the equation.

Conclusion

Solving exponential equations can be a challenging task, but with practice and patience, you can become proficient in solving them. Remember to isolate the exponential term, use properties of exponents, and evaluate the solutions correctly. With these tips and techniques, you can solve exponential equations with confidence.

Final Tips

• Practice solving exponential equations regularly to build your skills and confidence.
• Use properties of exponents and logarithms to simplify the equation and isolate the variable.
• Evaluate the solutions correctly to ensure that you have found the correct solution.
• Use a calculator to check your work and ensure that your solution is correct.

By following these tips and techniques, you can become proficient in solving exponential equations and tackle even the most challenging problems with confidence.