Which Of The Following Is The Solution Of $5 E^{2 X} - 4 = 11$?$A. $x = \ln 3$B. \$x = \ln 27$[/tex\]C. $x = \frac{\ln 3}{2}$D. $x = \frac{3}{\ln 3}$
Introduction
Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of exponential functions. In this article, we will explore the solution to the equation $5 e^{2 x} - 4 = 11$ and examine the different options provided.
Understanding Exponential Functions
Exponential functions are a type of mathematical function that describes a relationship between two variables, typically denoted as x and y. The general form of an exponential function is $y = a^x$, where a is a positive constant and x is the variable. In the case of the equation $5 e^{2 x} - 4 = 11$, the base of the exponential function is e, which is a mathematical constant approximately equal to 2.71828.
Solving the Equation
To solve the equation $5 e^{2 x} - 4 = 11$, we need to isolate the exponential term. The first step is to add 4 to both sides of the equation, which gives us $5 e^{2 x} = 15$. Next, we divide both sides of the equation by 5, resulting in $e^{2 x} = 3$.
Using Properties of Exponential Functions
Now that we have isolated the exponential term, we can use the properties of exponential functions to solve for x. One of the key properties of exponential functions is that $e^x$ is equal to the natural logarithm of x, denoted as $\ln x$. Therefore, we can rewrite the equation $e^{2 x} = 3$ as $2 x = \ln 3$.
Solving for x
To solve for x, we need to isolate x on one side of the equation. We can do this by dividing both sides of the equation by 2, resulting in $x = \frac{\ln 3}{2}$.
Evaluating the Options
Now that we have solved the equation, we can evaluate the options provided.
- Option A: $x = \ln 3$
- Option B: $x = \ln 27$
- Option C: $x = \frac{\ln 3}{2}$
- Option D: $x = \frac{3}{\ln 3}$
Based on our solution, we can see that option C is the correct answer.
Conclusion
In conclusion, solving exponential equations requires a deep understanding of the properties of exponential functions. By isolating the exponential term and using the properties of exponential functions, we can solve for x. In this article, we solved the equation $5 e^{2 x} - 4 = 11$ and evaluated the options provided. We found that option C, $x = \frac{\ln 3}{2}$, is the correct answer.
Additional Tips and Tricks
- When solving exponential equations, it's essential to isolate the exponential term.
- Use the properties of exponential functions to simplify the equation.
- Be careful when evaluating the options, as the correct answer may not be immediately apparent.
Common Mistakes to Avoid
- Failing to isolate the exponential term.
- Not using the properties of exponential functions to simplify the equation.
- Not carefully evaluating the options.
Real-World Applications
Exponential equations have numerous real-world applications, including:
- Modeling population growth and decline.
- Describing the behavior of electrical circuits.
- Analyzing the spread of diseases.
Final Thoughts
Frequently Asked Questions
Q: What is an exponential equation?
A: An exponential equation is a mathematical equation that involves an exponential function, which is a function of the form $y = a^x$, where a is a positive constant and x is the variable.
Q: How do I solve an exponential equation?
A: To solve an exponential equation, you need to isolate the exponential term and use the properties of exponential functions to simplify the equation. You can use the following steps:
- Isolate the exponential term.
- Use the properties of exponential functions to simplify the equation.
- Solve for x.
Q: What are some common properties of exponential functions?
A: Some common properties of exponential functions include:
-
e^x$ is equal to the natural logarithm of x, denoted as $\ln x$.
-
a^x$ is equal to $e^{x \ln a}$.
-
a^x$ is equal to $\frac{e^{x \ln a}}{e^0}$.
Q: How do I evaluate the options in an exponential equation?
A: To evaluate the options in an exponential equation, you need to substitute each option into the equation and check if it is true. You can use the following steps:
- Substitute each option into the equation.
- Check if the equation is true.
- Select the option that makes the equation true.
Q: What are some common mistakes to avoid when solving exponential equations?
A: Some common mistakes to avoid when solving exponential equations include:
- Failing to isolate the exponential term.
- Not using the properties of exponential functions to simplify the equation.
- Not carefully evaluating the options.
Q: How do I apply exponential equations in real-world situations?
A: Exponential equations have numerous real-world applications, including:
- Modeling population growth and decline.
- Describing the behavior of electrical circuits.
- Analyzing the spread of diseases.
Q: What are some examples of exponential equations?
A: Some examples of exponential equations include:
Q: How do I graph exponential functions?
A: To graph exponential functions, you need to use a graphing calculator or a computer program. You can also use the following steps:
- Plot the points on the graph.
- Draw a smooth curve through the points.
- Label the axes and the graph.
Q: What are some common applications of exponential functions?
A: Some common applications of exponential functions include:
- Modeling population growth and decline.
- Describing the behavior of electrical circuits.
- Analyzing the spread of diseases.
Q: How do I use exponential functions to model real-world situations?
A: To use exponential functions to model real-world situations, you need to:
- Identify the variables and the relationships between them.
- Choose an exponential function that models the situation.
- Use the function to make predictions and analyze the data.
Q: What are some common challenges when working with exponential functions?
A: Some common challenges when working with exponential functions include:
- Difficulty in isolating the exponential term.
- Difficulty in using the properties of exponential functions to simplify the equation.
- Difficulty in evaluating the options.
Conclusion
Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of exponential functions. By isolating the exponential term and using the properties of exponential functions, we can solve for x. In this article, we answered some frequently asked questions about exponential equations and provided tips and tricks for solving them. We also discussed some common mistakes to avoid and real-world applications of exponential equations.