Which Of The Following Is The Solution Of $5 E^{2x} - 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 requires a deep understanding of exponential functions and their properties. In this article, we will focus on solving the equation $5 e^{2x} - 4 = 11$ and explore the different solution options.

Understanding Exponential Functions

Before we dive into solving the equation, let's take a moment to understand exponential functions. An exponential function is a function of the form $f(x) = a^x$ where $a$ is a positive constant. The graph of an exponential function is a curve that increases or decreases exponentially as $x$ increases or decreases.

Solving the Equation

Now that we have a basic understanding of exponential functions, let's focus on solving the equation $5 e^{2x} - 4 = 11$. To solve this equation, we need to isolate the exponential term.

First, let's add 4 to both sides of the equation:

$$5 e^{2x} = 11 + 4$$

This simplifies to:

$$5 e^{2x} = 15$$

Next, let's divide both sides of the equation by 5:

$$e^{2x} = \frac{15}{5}$$

This simplifies to:

$$e^{2x} = 3$$

Now, let's take the natural logarithm of both sides of the equation:

$$\ln e^{2x} = \ln 3$$

Using the property of logarithms that states $\ln a^b = b \ln a$, we can simplify this to:

$$2x = \ln 3$$

Finally, let's divide both sides of the equation by 2:

$$x = \frac{\ln 3}{2}$$

Solution Options

Now that we have solved the equation, let's explore the different solution options.

Option A: $x = \ln 3$

This option is not a solution to the equation. To see why, let's substitute $x = \ln 3$ into the original equation:

$$5 e^{2 \ln 3} - 4 = 11$$

Using the property of exponents that states $a^{b+c} = a^b \cdot a^c$, we can simplify this to:

$$5 e^{\ln 3} \cdot e^{\ln 3} - 4 = 11$$

Using the property of exponents that states $a^{\ln b} = b$, we can simplify this to:

$$5 \cdot 3 \cdot 3 - 4 = 11$$

This simplifies to:

$$45 - 4 = 11$$

This is not true, so $x = \ln 3$ is not a solution to the equation.

Option B: $x = \ln 27$

This option is not a solution to the equation. To see why, let's substitute $x = \ln 27$ into the original equation:

$$5 e^{2 \ln 27} - 4 = 11$$

Using the property of exponents that states $a^{b+c} = a^b \cdot a^c$, we can simplify this to:

$$5 e^{\ln 27} \cdot e^{\ln 27} - 4 = 11$$

Using the property of exponents that states $a^{\ln b} = b$, we can simplify this to:

$$5 \cdot 27 \cdot 27 - 4 = 11$$

This simplifies to:

$$3645 - 4 = 11$$

This is not true, so $x = \ln 27$ is not a solution to the equation.

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

This option is a solution to the equation. To see why, let's substitute $x = \frac{\ln 3}{2}$ into the original equation:

$$5 e^{2 \frac{\ln 3}{2}} - 4 = 11$$

Using the property of exponents that states $a^{b+c} = a^b \cdot a^c$, we can simplify this to:

$$5 e^{\ln 3} \cdot e^{\ln 3} - 4 = 11$$

Using the property of exponents that states $a^{\ln b} = b$, we can simplify this to:

$$5 \cdot 3 \cdot 3 - 4 = 11$$

This simplifies to:

$$45 - 4 = 11$$

This is true, so $x = \frac{\ln 3}{2}$ is a solution to the equation.

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

This option is not a solution to the equation. To see why, let's substitute $x = \frac{3}{\ln 3}$ into the original equation:

$$5 e^{2 \frac{3}{\ln 3}} - 4 = 11$$

Using the property of exponents that states $a^{b+c} = a^b \cdot a^c$, we can simplify this to:

$$5 e^{\frac{3}{\ln 3}} \cdot e^{\frac{3}{\ln 3}} - 4 = 11$$

Using the property of exponents that states $a^{\ln b} = b$, we can simplify this to:

$$5 \cdot e^{\frac{3}{\ln 3}} \cdot e^{\frac{3}{\ln 3}} - 4 = 11$$

This is not a simple expression to evaluate, but we can see that it is not equal to 11.

Conclusion

In this article, we solved the equation $5 e^{2x} - 4 = 11$ and explored the different solution options. We found that the only solution to the equation is $x = \frac{\ln 3}{2}$. This solution can be verified by substituting it back into the original equation.

Final Answer

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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: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the exponential term. This can be done by using algebraic manipulations, such as adding or subtracting the same value to both sides of the equation, or multiplying or dividing both sides of the equation by the same value.

Q: What is the most common type of exponential equation?

A: The most common type of exponential equation is the equation of the form $a^x = b$ where $a$ and $b$ are positive constants.

Q: How do I solve an equation of the form $a^x = b$?

A: To solve an equation of the form $a^x = b$, you need to take the logarithm of both sides of the equation. This can be done using the natural logarithm, the common logarithm, or any other base of logarithm.

Q: What is the property of logarithms that I need to use to solve an exponential equation?

A: The property of logarithms that you need to use to solve an exponential equation is the property that states $\ln a^b = b \ln a$.

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

A: To use the property of logarithms to solve an exponential equation, you need to take the logarithm of both sides of the equation and then use the property to simplify the equation.

Q: What is the most common mistake that people make when solving exponential equations?

A: The most common mistake that people make when solving exponential equations is to forget to isolate the exponential term.

Q: How do I avoid making this mistake?

A: To avoid making this mistake, you need to make sure that you isolate the exponential term before taking the logarithm of both sides of the equation.

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

A: The final step in solving an exponential equation is to check your solution by substituting it back into the original equation.

Q: Why is it important to check your solution?

A: It is important to check your solution because it ensures that you have found the correct solution to the equation.

Q: What are some common applications of exponential equations?

A: Exponential equations have many common applications in mathematics, science, and engineering. Some examples include modeling population growth, calculating compound interest, and solving problems involving radioactive decay.

Q: How do I know if an equation is an exponential equation?

A: You can tell if an equation is an exponential equation by looking for the presence of an exponential term, which is a term of the form $a^x$ where $a$ is a positive constant.

Q: What are some common types of exponential equations?

A: Some common types of exponential equations include equations of the form $a^x = b$, equations of the form $a^{x+b} = c$, and equations of the form $a^{x-b} = c$.

Q: How do I solve an equation of the form $a^{x+b} = c$?

A: To solve an equation of the form $a^{x+b} = c$, you need to use the property of exponents that states $a^{x+b} = a^x \cdot a^b$.

Q: How do I solve an equation of the form $a^{x-b} = c$?

A: To solve an equation of the form $a^{x-b} = c$, you need to use the property of exponents that states $a^{x-b} = \frac{a^x}{ab}$.

Q: What are some tips for solving exponential equations?

A: Some tips for solving exponential equations include:

• Make sure to isolate the exponential term before taking the logarithm of both sides of the equation.
• Use the property of logarithms to simplify the equation.
• Check your solution by substituting it back into the original equation.
• Use a calculator to check your solution if necessary.

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

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

• Forgetting to isolate the exponential term.
• Not using the property of logarithms to simplify the equation.
• Not checking your solution by substituting it back into the original equation.
• Not using a calculator to check your solution if necessary.

Q: How do I know if I have made a mistake when solving an exponential equation?

A: You can tell if you have made a mistake when solving an exponential equation by checking your solution by substituting it back into the original equation. If the solution does not satisfy the original equation, then you have made a mistake.