Solve The Following Exponential Equation. Express Your Answer As Both An Exact Expression And A Decimal Approximation Rounded To Two Decimal Places.${ 3 E^{x-1} = 77 }$

Introduction

Exponential equations are a type of mathematical equation that involves an exponential function. These equations can be challenging to solve, but with the right techniques and strategies, they can be tackled with ease. In this article, we will focus on solving the exponential equation ${3e^{x-1} = 77}$ and express the solution as both an exact expression and a decimal approximation rounded to two decimal places.

Understanding Exponential Functions

Before we dive into solving the equation, let's take a moment to understand what exponential functions are. An exponential function is a function that can be written in the form ${f(x) = ab^x}$, where ${a}$ and ${b}$ are constants, and ${b}$ is the base of the exponential function. In the case of the equation ${3e^{x-1} = 77}$, the base of the exponential function is ${e}$, which is a mathematical constant approximately equal to 2.71828.

Solving the Exponential Equation

To solve the equation ${3e^{x-1} = 77}$, we need to isolate the exponential term. We can do this by dividing both sides of the equation by 3, which gives us:

${e^{x-1} = \frac{77}{3}}$

Next, we can take the natural logarithm (ln) of both sides of the equation to get rid of the exponential term. The natural logarithm is the inverse of the exponential function, and it is denoted by the symbol ln. Taking the natural logarithm of both sides of the equation gives us:

${x-1 = \ln\left(\frac{77}{3}\right)}$

Now, we can add 1 to both sides of the equation to isolate the variable x:

${x = \ln\left(\frac{77}{3}\right) + 1}$

Exact Expression

The exact expression for the solution is:

${x = \ln\left(\frac{77}{3}\right) + 1}$

This expression cannot be simplified further, and it is the exact solution to the equation.

Decimal Approximation

To find the decimal approximation of the solution, we can use a calculator to evaluate the natural logarithm of ${\frac{77}{3}}$. The natural logarithm of ${\frac{77}{3}}$ is approximately 4.287. Adding 1 to this value gives us:

${x \approx 4.287 + 1 = 5.287}$

Rounding this value to two decimal places gives us:

${x \approx 5.29}$

Conclusion

In this article, we solved the exponential equation ${3e^{x-1} = 77}$ and expressed the solution as both an exact expression and a decimal approximation rounded to two decimal places. The exact expression for the solution is ${x = \ln\left(\frac{77}{3}\right) + 1}$, and the decimal approximation is ${x \approx 5.29}$. We hope that this article has provided a clear and concise guide to solving exponential equations, and we encourage readers to practice solving these types of equations to become more proficient.

Additional Resources

For readers who want to learn more about exponential functions and how to solve exponential equations, we recommend checking out the following resources:

• Khan Academy: Exponential Functions
• Mathway: Exponential Equations
• Wolfram Alpha: Exponential Functions

These resources provide a wealth of information and practice problems to help readers become more confident in their ability to solve exponential equations.

Final Thoughts

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Q: What is an exponential equation?

A: An exponential equation is a type of mathematical equation that involves an exponential function. Exponential functions are functions that can be written in the form ${f(x) = ab^x}$, where ${a}$ and ${b}$ are constants, and ${b}$ is the base of the exponential function.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the exponential term. You can do this by dividing both sides of the equation by the coefficient of the exponential term, and then taking the natural logarithm (ln) of both sides of the equation.

Q: What is the natural logarithm (ln)?

A: The natural logarithm (ln) is the inverse of the exponential function. It is denoted by the symbol ln, and it is used to solve exponential equations.

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

A: To use the natural logarithm (ln) to solve an exponential equation, you need to take the natural logarithm of both sides of the equation. This will eliminate the exponential term and allow you to 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. Logarithmic functions are the inverse of exponential functions.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to isolate the logarithmic term. You can do this by using the properties of logarithms, such as the product rule and the quotient rule.

Q: What are the product rule and the quotient rule for logarithms?

A: The product rule for logarithms states that ${\log_b (xy) = \log_b x + \log_b y}$, while the quotient rule for logarithms states that ${\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y}$.

Q: How do I use the product rule and the quotient rule for logarithms to solve a logarithmic equation?

A: To use the product rule and the quotient rule for logarithms to solve a logarithmic equation, you need to apply these rules to the logarithmic term and then simplify the equation.

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

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

• Not isolating the exponential or logarithmic term
• Not using the correct properties of logarithms
• Not simplifying the equation correctly
• Not checking the solution for extraneous solutions

Q: How can I practice solving exponential and logarithmic equations?

A: You can practice solving exponential and logarithmic equations by working through example problems and exercises. You can also use online resources, such as Khan Academy and Mathway, to practice solving these types of equations.

Conclusion

In this article, we answered some frequently asked questions about exponential and logarithmic equations. We hope that this article has provided a clear and concise guide to solving these types of equations, and we encourage readers to practice solving them to become more proficient.