Solve The Equation. Round To The Nearest Ten-thousandth.${ 5 {x 2-3} = 72 }${$ X = \pm $}$ ${ \square }$

Introduction

In this article, we will delve into the world of mathematics and explore a complex equation that requires a deep understanding of exponential functions and logarithms. The equation in question is $5^{x^2-3} = 72$, and our goal is to solve for the value of x. We will use various mathematical techniques and tools to uncover the solution, and we will also discuss the importance of precision in mathematical calculations.

Understanding the Equation

The given equation is $5^{x^2-3} = 72$. To begin solving this equation, we need to understand the properties of exponential functions and logarithms. The exponential function $5^x$ is a function that raises 5 to the power of x, where x is a real number. The logarithmic function, on the other hand, is the inverse of the exponential function. In this case, we are dealing with a logarithmic equation, where the base is 5 and the result is 72.

Using Logarithms to Solve the Equation

One way to solve this equation is to use logarithms. We can take the logarithm of both sides of the equation, which will allow us to eliminate the exponent. Using the logarithmic property $\log_a{b^c} = c \cdot \log_a{b}$, we can rewrite the equation as:

$$\log_5{72} = x^2 - 3$$

Applying the Change of Base Formula

To solve for x, we need to isolate x^2. We can do this by adding 3 to both sides of the equation, which gives us:

$$\log_5{72} + 3 = x^2$$

However, we still need to find the value of $\log_5{72}$. To do this, we can use the change of base formula, which states that $\log_a{b} = \frac{\log_c{b}}{\log_c{a}}$. In this case, we can choose base 10 as our new base, which gives us:

$$\log_5{72} = \frac{\log_{10}{72}}{\log_{10}{5}}$$

Evaluating the Logarithmic Expression

Using a calculator, we can evaluate the logarithmic expression:

$$\frac{\log_{10}{72}}{\log_{10}{5}} \approx 1.877$$

Solving for x

Now that we have the value of $\log_5{72}$, we can substitute it back into the equation:

$$1.877 + 3 = x^2$$

Simplifying the equation, we get:

$$4.877 = x^2$$

Taking the Square Root

To solve for x, we need to take the square root of both sides of the equation. This gives us:

$$x = \pm \sqrt{4.877}$$

Evaluating the Square Root

Using a calculator, we can evaluate the square root:

$$\sqrt{4.877} \approx 2.21$$

The Final Solution

Therefore, the final solution to the equation is:

$$x = \pm 2.21$$

Discussion

In this article, we have demonstrated how to solve a complex equation using logarithms and the change of base formula. We have also discussed the importance of precision in mathematical calculations, as small errors can lead to significant differences in the final solution. The equation $5^{x^2-3} = 72$ is a classic example of a logarithmic equation, and solving it requires a deep understanding of exponential functions and logarithms.

Conclusion

In conclusion, solving the equation $5^{x^2-3} = 72$ requires a combination of mathematical techniques and tools, including logarithms and the change of base formula. By following the steps outlined in this article, we have been able to uncover the value of x, which is $\pm 2.21$. This solution demonstrates the power of mathematics in solving complex problems and highlights the importance of precision in mathematical calculations.

Additional Resources

For those who want to learn more about logarithms and exponential functions, there are many online resources available, including:

• Khan Academy: Logarithms and Exponential Functions
• Mathway: Logarithmic Equations
• Wolfram Alpha: Logarithmic Equations

These resources provide a wealth of information and examples to help you understand and apply logarithmic and exponential functions in a variety of contexts.

Final Thoughts

Solving the equation $5^{x^2-3} = 72$ is a challenging problem that requires a deep understanding of mathematical concepts and techniques. By following the steps outlined in this article, we have been able to uncover the value of x, which is $\pm 2.21$. This solution demonstrates the power of mathematics in solving complex problems and highlights the importance of precision in mathematical calculations. Whether you are a student, teacher, or simply someone who is interested in mathematics, this article provides a valuable resource for learning and understanding logarithmic and exponential functions.

Introduction

In our previous article, we explored the equation $5^{x^2-3} = 72$ and solved for the value of x. In this article, we will answer some of the most frequently asked questions about the equation and its solution.

Q: What is the equation $5^{x^2-3} = 72$ trying to tell us?

A: The equation $5^{x^2-3} = 72$ is a logarithmic equation that involves an exponential function. The equation is trying to tell us that the value of $5^{x^2-3}$ is equal to 72.

Q: How do we solve the equation $5^{x^2-3} = 72$?

A: To solve the equation $5^{x^2-3} = 72$, we can use logarithms and the change of base formula. We can take the logarithm of both sides of the equation, which will allow us to eliminate the exponent.

Q: What is the change of base formula?

A: The change of base formula is a mathematical formula that allows us to change the base of a logarithm. The formula is $\log_a{b} = \frac{\log_c{b}}{\log_c{a}}$, where a, b, and c are positive real numbers.

Q: How do we use the change of base formula to solve the equation $5^{x^2-3} = 72$?

A: To use the change of base formula to solve the equation $5^{x^2-3} = 72$, we can choose a new base, such as base 10, and rewrite the equation as $\log_{10}{72} = (x^2-3) \cdot \log_{10}{5}$.

Q: What is the value of $\log_{10}{72}$?

A: The value of $\log_{10}{72}$ is approximately 1.877.

Q: How do we solve for x in the equation $5^{x^2-3} = 72$?

A: To solve for x in the equation $5^{x^2-3} = 72$, we can add 3 to both sides of the equation, which gives us $\log_{10}{72} + 3 = x^2$. We can then take the square root of both sides of the equation to solve for x.

Q: What is the value of x in the equation $5^{x^2-3} = 72$?

A: The value of x in the equation $5^{x^2-3} = 72$ is approximately $\pm 2.21$.

Q: Why is it important to be precise in mathematical calculations?

A: It is essential to be precise in mathematical calculations because small errors can lead to significant differences in the final solution. In the case of the equation $5^{x^2-3} = 72$, a small error in the calculation of $\log_{10}{72}$ can result in a significantly different value of x.

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

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

• Not using the correct base for the logarithm
• Not using the change of base formula correctly
• Not checking the domain of the logarithmic function
• Not being precise in mathematical calculations

Q: How can I practice solving logarithmic equations?

A: You can practice solving logarithmic equations by working through examples and exercises in a textbook or online resource. You can also try solving logarithmic equations on your own, using a calculator or computer program to check your work.

Q: What are some real-world applications of logarithmic equations?

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

• Modeling population growth and decay
• Analyzing financial data and predicting stock prices
• Understanding the behavior of complex systems and networks
• Solving problems in physics, engineering, and computer science

Conclusion

In this article, we have answered some of the most frequently asked questions about the equation $5^{x^2-3} = 72$ and its solution. We have discussed the importance of precision in mathematical calculations and provided tips for avoiding common mistakes when solving logarithmic equations. Whether you are a student, teacher, or simply someone who is interested in mathematics, this article provides a valuable resource for learning and understanding logarithmic and exponential functions.