Solve For N N N :${ 10 - 60e^{20n+2} = -4126.4 }$Write Your Answer As A Decimal Rounded To Four Decimal Places. N ≈ □ N \approx \square N ≈ □

Introduction

Exponential equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving a specific exponential equation involving the variable $n$. The equation is given as:

$$10 - 60e^{20n+2} = -4126.4$$

Our goal is to isolate the variable $n$ and find its value as a decimal rounded to four decimal places.

Step 1: Isolate the Exponential Term

The first step in solving this equation is to isolate the exponential term. To do this, we need to move the constant term to the other side of the equation.

$$-60e^{20n+2} = -4136.4 - 10$$

$$-60e^{20n+2} = -4146.4$$

Now, we can divide both sides of the equation by -60 to isolate the exponential term.

$$e^{20n+2} = \frac{-4146.4}{-60}$$

$$e^{20n+2} = 69.1$$

Step 2: Take the Natural Logarithm

The next step is to take the natural logarithm (ln) of both sides of the equation. This will help us eliminate the exponential term.

$$\ln(e^{20n+2}) = \ln(69.1)$$

Using the property of logarithms that states $\ln(e^x) = x$, we can simplify the left-hand side of the equation.

$$20n + 2 = \ln(69.1)$$

Step 3: Solve for $n$

Now that we have isolated the term with the variable $n$, we can solve for $n$. To do this, we need to subtract 2 from both sides of the equation.

$$20n = \ln(69.1) - 2$$

Next, we can divide both sides of the equation by 20 to find the value of $n$.

$$n = \frac{\ln(69.1) - 2}{20}$$

Step 4: Calculate the Value of $n$

Now that we have the equation for $n$, we can calculate its value. Using a calculator, we can find the value of $\ln(69.1)$ and then plug it into the equation.

$$n = \frac{\ln(69.1) - 2}{20}$$

$$n = \frac{4.734 - 2}{20}$$

$$n = \frac{2.734}{20}$$

$$n = 0.1367$$

Conclusion

In this article, we solved an exponential equation involving the variable $n$. We started by isolating the exponential term, took the natural logarithm of both sides, and then solved for $n$. The final value of $n$ is approximately 0.1367, rounded to four decimal places.

Tips and Variations

• When solving exponential equations, it's essential to isolate the exponential term first.
• Taking the natural logarithm of both sides can help eliminate the exponential term.
• Be careful when using logarithms, as the base of the logarithm must be the same on both sides of the equation.
• Exponential equations can be challenging to solve, but with practice and patience, you can become proficient in solving them.

Common Mistakes to Avoid

• Not isolating the exponential term first.
• Not taking the natural logarithm of both sides.
• Using the wrong base for the logarithm.
• Not rounding the final answer to the correct number of decimal places.

Real-World Applications

Exponential equations have many real-world applications, including:

• Modeling population growth and decline.
• Describing chemical reactions.
• Analyzing financial data.
• Predicting weather patterns.

Introduction

In our previous article, we solved an exponential equation involving the variable $n$. We received many questions from readers who wanted to learn more about solving exponential equations. In this article, we will answer some of the most frequently asked questions about solving exponential equations.

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential term, which is a term that is raised to a power. Exponential equations can be written in the form $a^x = b$, where $a$ is the base, $x$ is the exponent, and $b$ is the result.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the exponential term first. This can be done by moving the constant term to the other side of the equation. Then, you can take the natural logarithm of both sides to eliminate the exponential term. Finally, you can solve for the variable.

Q: What is the natural logarithm?

A: The natural logarithm, denoted by $\ln$, is the logarithm of a number to the base $e$. It is used to eliminate the exponential term in an exponential equation.

Q: How do I use the natural logarithm to solve an exponential equation?

A: To use the natural logarithm to solve an exponential equation, you need to take the natural logarithm of both sides of the equation. This will help you eliminate the exponential term. For example, if you have the equation $e^x = 5$, you can take the natural logarithm of both sides to get:

$$\ln(e^x) = \ln(5)$$

Using the property of logarithms that states $\ln(e^x) = x$, you can simplify the left-hand side of the equation to get:

$$x = \ln(5)$$

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

A: An exponential equation is an equation that involves an exponential term, while a logarithmic equation is an equation that involves a logarithmic term. Exponential equations can be written in the form $a^x = b$, while logarithmic equations can be written in the form $\log_a(b) = x$.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to isolate the logarithmic term first. This can be done by moving the constant term to the other side of the equation. Then, you can use the definition of a logarithm to rewrite the equation in exponential form. Finally, you can solve for the variable.

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 first.
• Not taking the natural logarithm of both sides.
• Using the wrong base for the logarithm.
• Not rounding the final answer to the correct number of decimal places.

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

A: Exponential equations can be used to model population growth and decline, describe chemical reactions, analyze financial data, and predict weather patterns. By understanding how to solve exponential equations, you can apply this knowledge to a wide range of real-world problems.

Conclusion

In this article, we answered some of the most frequently asked questions about solving exponential equations. We hope that this article has been helpful in clarifying any confusion you may have had about solving exponential equations. If you have any further questions, please don't hesitate to ask.

Tips and Variations

• When solving exponential equations, it's essential to isolate the exponential term first.
• Taking the natural logarithm of both sides can help eliminate the exponential term.
• Be careful when using logarithms, as the base of the logarithm must be the same on both sides of the equation.
• Exponential equations can be challenging to solve, but with practice and patience, you can become proficient in solving them.

Common Mistakes to Avoid

• Not isolating the exponential term first.
• Not taking the natural logarithm of both sides.
• Using the wrong base for the logarithm.
• Not rounding the final answer to the correct number of decimal places.

Real-World Applications

Exponential equations have many real-world applications, including:

• Modeling population growth and decline.
• Describing chemical reactions.
• Analyzing financial data.
• Predicting weather patterns.

By understanding how to solve exponential equations, you can apply this knowledge to a wide range of real-world problems.