Solve $e^{2x} = 40$. Round To The Thousandths Place.A. $x = 0.801$ B. $x = 3.689$ C. $x = 1.602$ D. $x = 1.844$

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Introduction

Exponential equations are a fundamental concept in mathematics, and solving them is crucial in various fields such as physics, engineering, and economics. In this article, we will focus on solving the equation $e^{2x} = 40$ and round the solution to the thousandths place.

What are Exponential Equations?

Exponential equations are equations that involve an exponential function, which is a function of the form $f(x) = a^x$, where $a$ is a positive constant. The exponential function is a fundamental concept in mathematics and has numerous applications in various fields.

The Equation $e^{2x} = 40$

The equation $e^{2x} = 40$ is an exponential equation that involves the base $e$, which is approximately equal to $2.71828$. To solve this equation, we need to isolate the variable $x$.

Step 1: Take the Natural Logarithm of Both Sides

To solve the equation $e^{2x} = 40$, we can take the natural logarithm of both sides. The natural logarithm is the logarithm to the base $e$, and it is denoted by $\ln(x)$. Taking the natural logarithm of both sides of the equation gives us:

ln⁡(e2x)=ln⁡(40)\ln(e^{2x}) = \ln(40)

Step 2: Simplify the Equation

Using the property of logarithms that states $\ln(a^b) = b\ln(a)$, we can simplify the equation:

2xln⁡(e)=ln⁡(40)2x\ln(e) = \ln(40)

Since $\ln(e) = 1$, we can simplify the equation further:

2x=ln⁡(40)2x = \ln(40)

Step 3: Solve for x

To solve for $x$, we can divide both sides of the equation by $2$:

x=ln⁡(40)2x = \frac{\ln(40)}{2}

Step 4: Calculate the Value of x

To calculate the value of $x$, we can use a calculator to evaluate the expression $\frac{\ln(40)}{2}$. Plugging in the value of $\ln(40)$, which is approximately equal to $3.6889$, we get:

x=3.68892x = \frac{3.6889}{2}

x=1.84445x = 1.84445

Rounding the Solution to the Thousandths Place

To round the solution to the thousandths place, we can round the value of $x$ to three decimal places:

x=1.844x = 1.844

Conclusion

In this article, we solved the equation $e^{2x} = 40$ and rounded the solution to the thousandths place. The solution is $x = 1.844$.

Comparison with Other Options

To compare our solution with other options, let's evaluate the expressions $x = 0.801$, $x = 3.689$, and $x = 1.602$.

  • x = 0.801$: This value is not equal to the solution we obtained.

  • x = 3.689$: This value is not equal to the solution we obtained.

  • x = 1.602$: This value is not equal to the solution we obtained.

Conclusion

In conclusion, the correct solution to the equation $e^{2x} = 40$ is $x = 1.844$.

Final Answer

Introduction

In our previous article, we solved the equation $e^{2x} = 40$ and rounded the solution to the thousandths place. In this article, we will provide a Q&A guide to help you understand the concept of exponential equations and how to solve them.

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 is the Base of an Exponential Function?

A: The base of an exponential function is the constant $a$ in the function $f(x) = a^x$. For example, in the function $f(x) = 2^x$, the base is $2$.

Q: What is the Natural Logarithm?

A: The natural logarithm is the logarithm to the base $e$, where $e$ is approximately equal to $2.71828$. It is denoted by $\ln(x)$.

Q: How Do I Solve an Exponential Equation?

A: To solve an exponential equation, you can use the following steps:

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

Q: What is the Difference Between an Exponential Equation and a Logarithmic Equation?

A: An exponential equation is an equation that involves an exponential function, while a logarithmic equation is an equation that involves a logarithmic function. For example, the equation $e^{2x} = 40$ is an exponential equation, while the equation $\ln(x) = 2$ is a logarithmic equation.

Q: Can I Use a Calculator to Solve Exponential Equations?

A: Yes, you can use a calculator to solve exponential equations. However, it's always a good idea to understand the concept behind the solution and to check your work.

Q: What are Some Common Exponential Equations?

A: Some common exponential equations include:

  • e2x=40e^{2x} = 40

  • 2x=162^x = 16

  • 3x=273^x = 27

Q: How Do I Round a Solution to the Thousandths Place?

A: To round a solution to the thousandths place, you can use the following steps:

  1. Evaluate the expression to find the solution.
  2. Look at the thousandths place digit.
  3. If the digit is less than $5$, round down.
  4. If the digit is $5$ or greater, round up.

Conclusion

In conclusion, solving exponential equations requires a good understanding of the concept of exponential functions and logarithmic functions. By following the steps outlined in this article, you can solve exponential equations and round your solutions to the thousandths place.

Final Tips

  • Always check your work when solving exponential equations.
  • Use a calculator to evaluate expressions and check your work.
  • Practice solving exponential equations to become more confident in your abilities.

Common Mistakes to Avoid

  • Not taking the natural logarithm of both sides of the equation.
  • Not simplifying the equation using the properties of logarithms.
  • Not solving for the variable.

Conclusion

In conclusion, solving exponential equations is a crucial concept in mathematics, and it requires a good understanding of the concept of exponential functions and logarithmic functions. By following the steps outlined in this article and avoiding common mistakes, you can become more confident in your abilities to solve exponential equations.