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

by ADMIN 143 views

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of exponential functions and logarithms. In this article, we will focus on solving the equation $5 e^{2 x} - 4 = 11$ and explore the different solution options provided.

Understanding Exponential Equations

Exponential equations involve an exponential function, which is a function of the form f(x)=axf(x) = a^x, where aa is a positive constant. The exponential function exe^x is a special case of this form, where a=ea = e. Exponential equations can be solved using logarithms, which are the inverse operation of exponentiation.

Solving the Equation

To solve the equation $5 e^{2 x} - 4 = 11$, we need to isolate the exponential term. We can start by adding 4 to both sides of the equation:

5e2x=155 e^{2 x} = 15

Next, we can divide both sides of the equation by 5:

e2x=3e^{2 x} = 3

Now, we can take the natural logarithm of both sides of the equation to eliminate the exponential term:

2x=ln⁡32 x = \ln 3

Finally, we can divide both sides of the equation by 2 to solve for xx:

x=ln⁡32x = \frac{\ln 3}{2}

Evaluating the Solution Options

Now that we have solved the equation, let's evaluate the solution options provided:

A. x=ln⁡3x = \ln 3 B. x=ln⁡27x = \ln 27 C. x=ln⁡32x = \frac{\ln 3}{2} D. x=3ln⁡3x = \frac{3}{\ln 3}

Option A is incorrect because it does not match the solution we obtained. Option B is also incorrect because it is not equal to the solution we obtained. Option D is incorrect because it is not equal to the solution we obtained.

Conclusion

In conclusion, the solution to the equation $5 e^{2 x} - 4 = 11$ is x=ln⁡32x = \frac{\ln 3}{2}. This solution was obtained by isolating the exponential term, taking the natural logarithm of both sides, and solving for xx. We also evaluated the solution options provided and found that only option C matches the solution we obtained.

Additional Tips and Tricks

When solving exponential equations, it's essential to remember the following tips and tricks:

  • Isolate the exponential term by adding or subtracting the same value from both sides of the equation.
  • Use logarithms to eliminate the exponential term.
  • Simplify the equation by combining like terms.
  • Check your solution by plugging it back into the original equation.

By following these tips and tricks, you can solve exponential equations with confidence and accuracy.

Common Mistakes to Avoid

When solving exponential equations, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Not isolating the exponential term.
  • Not using logarithms to eliminate the exponential term.
  • Not simplifying the equation by combining like terms.
  • Not checking your solution by plugging it back into the original equation.

By avoiding these common mistakes, you can ensure that your solution is accurate and reliable.

Real-World Applications

Exponential equations have many real-world applications, including:

  • Modeling population growth and decline.
  • Calculating compound interest.
  • Analyzing chemical reactions.
  • Predicting stock prices.

By understanding how to solve exponential equations, you can apply this knowledge to real-world problems and make informed decisions.

Conclusion

Frequently Asked Questions

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)=axf(x) = a^x, where aa is a positive constant. Exponential equations can be solved using logarithms, which are the inverse operation of exponentiation.

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 solve for xx. Here's a step-by-step guide:

  1. Isolate the exponential term by adding or subtracting the same value from both sides of the equation.
  2. Take the natural logarithm of both sides of the equation.
  3. Simplify the equation by combining like terms.
  4. Solve for xx.

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

A: An exponential equation involves an exponential function, while a logarithmic equation involves a logarithmic function. Exponential equations can be solved using logarithms, while logarithmic equations can be solved using exponentiation.

Q: Can I use any type of logarithm to solve an exponential equation?

A: No, you can only use the natural logarithm (ln) to solve an exponential equation. The natural logarithm is the inverse operation of the exponential function exe^x.

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

A: To check your solution, plug it back into the original equation and simplify. If the equation is true, then your solution is correct.

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 logarithms to eliminate the exponential term.
  • Not simplifying the equation by combining like terms.
  • Not checking your solution by plugging it back into the original equation.

Q: How do I apply exponential equations to real-world problems?

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

  • Modeling population growth and decline.
  • Calculating compound interest.
  • Analyzing chemical reactions.
  • Predicting stock prices.

By understanding how to solve exponential equations, you can apply this knowledge to real-world problems and make informed decisions.

Q: Can I use exponential equations to solve problems involving negative exponents?

A: Yes, you can use exponential equations to solve problems involving negative exponents. To do this, you need to rewrite the negative exponent as a positive exponent by taking the reciprocal of the base.

Q: How do I simplify an exponential equation with multiple terms?

A: To simplify an exponential equation with multiple terms, combine like terms by adding or subtracting the exponents.

Q: Can I use exponential equations to solve problems involving complex numbers?

A: Yes, you can use exponential equations to solve problems involving complex numbers. To do this, you need to use the complex exponential function eixe^{ix}, where ii is the imaginary unit.

Conclusion

In conclusion, exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of exponential functions and logarithms. By following the steps outlined in this article, you can solve exponential equations with confidence and accuracy. Remember to isolate the exponential term, use logarithms to eliminate the exponential term, simplify the equation by combining like terms, and check your solution by plugging it back into the original equation. By avoiding common mistakes and applying your knowledge to real-world problems, you can become a master of exponential equations.