A Student Showed The Steps Below While Solving The Equation $14 = \log_5(2x - 3$\] By Graphing.1. Write A System Of Equations: $\[ Y_1 = 14 \\] $\[ Y_2 = \log_5(2x - 3) \\]2. Use The Change Of Base Formula To Rewrite The

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Introduction

In mathematics, solving equations involving logarithms can be a challenging task, especially when they are not in a standard form. One approach to solving such equations is by graphing, which involves using a graphical representation to find the solution. In this article, we will discuss a student's approach to solving the equation $14 = \log_5(2x - 3)$ by graphing.

Step 1: Write a System of Equations

The first step in solving the equation $14 = \log_5(2x - 3)$ by graphing is to write a system of equations. This involves creating two equations, one for each side of the equation.

• Equation 1: $y_1 = 14$
• Equation 2: $y_2 = \log_5(2x - 3)$

By writing these two equations, we can create a system of equations that can be graphed.

Step 2: Use the Change of Base Formula to Rewrite the Equation

The change of base formula is a mathematical formula that allows us to rewrite a logarithmic equation in terms of a different base. In this case, we can use the change of base formula to rewrite the equation $14 = \log_5(2x - 3)$ in terms of a common logarithm.

The change of base formula is given by:

$$\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$$

where $a$, $b$, and $c$ are positive real numbers.

Using this formula, we can rewrite the equation $14 = \log_5(2x - 3)$ as:

$$14 = \frac{\log(2x - 3)}{\log(5)}$$

This allows us to rewrite the equation in terms of a common logarithm, which can be easier to work with.

Step 3: Graph the System of Equations

Once we have written the system of equations and rewritten the equation using the change of base formula, we can graph the system of equations. This involves plotting the two equations on a coordinate plane and finding the point of intersection.

To graph the system of equations, we can use a graphing calculator or a computer program. We can also use a table of values to find the point of intersection.

Step 4: Find the Point of Intersection

Once we have graphed the system of equations, we can find the point of intersection by looking at the graph. The point of intersection is the point where the two equations intersect.

In this case, the point of intersection is the solution to the equation $14 = \log_5(2x - 3)$.

Conclusion

Solving equations involving logarithms can be a challenging task, especially when they are not in a standard form. One approach to solving such equations is by graphing, which involves using a graphical representation to find the solution. In this article, we discussed a student's approach to solving the equation $14 = \log_5(2x - 3)$ by graphing. We wrote a system of equations, used the change of base formula to rewrite the equation, graphed the system of equations, and found the point of intersection.

The Importance of Graphing in Mathematics

Graphing is an important tool in mathematics, as it allows us to visualize complex equations and find their solutions. By graphing a system of equations, we can find the point of intersection, which is the solution to the equation.

Real-World Applications of Graphing

Graphing has many real-world applications, including:

• Science: Graphing is used in science to model real-world phenomena, such as the motion of objects and the behavior of populations.
• Engineering: Graphing is used in engineering to design and optimize systems, such as bridges and buildings.
• Economics: Graphing is used in economics to model economic systems and make predictions about future trends.

Conclusion

In conclusion, graphing is an important tool in mathematics, as it allows us to visualize complex equations and find their solutions. By graphing a system of equations, we can find the point of intersection, which is the solution to the equation. Graphing has many real-world applications, including science, engineering, and economics.

References

• Change of Base Formula: The change of base formula is a mathematical formula that allows us to rewrite a logarithmic equation in terms of a different base.
• Graphing Calculator: A graphing calculator is a tool that allows us to graph equations and find their solutions.
• Computer Program: A computer program is a tool that allows us to graph equations and find their solutions.

Future Research Directions

Future research directions in graphing include:

• Developing new graphing algorithms: Developing new graphing algorithms that can handle complex equations and find their solutions more efficiently.
• Improving graphing software: Improving graphing software to make it more user-friendly and efficient.
• Applying graphing to real-world problems: Applying graphing to real-world problems, such as modeling population growth and predicting economic trends.

Conclusion

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Introduction

In our previous article, we discussed a student's approach to solving the equation $14 = \log_5(2x - 3)$ by graphing. We wrote a system of equations, used the change of base formula to rewrite the equation, graphed the system of equations, and found the point of intersection. In this article, we will answer some frequently asked questions about solving the equation $14 = \log_5(2x - 3)$ by graphing.

Q: What is the change of base formula?

A: The change of base formula is a mathematical formula that allows us to rewrite a logarithmic equation in terms of a different base. It is given by:

$$\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$$

where $a$, $b$, and $c$ are positive real numbers.

Q: Why do we need to use the change of base formula?

A: We need to use the change of base formula to rewrite the equation $14 = \log_5(2x - 3)$ in terms of a common logarithm. This allows us to graph the equation more easily and find the point of intersection.

Q: How do we graph the system of equations?

A: We can graph the system of equations using a graphing calculator or a computer program. We can also use a table of values to find the point of intersection.

Q: What is the point of intersection?

A: The point of intersection is the solution to the equation $14 = \log_5(2x - 3)$. It is the point where the two equations intersect.

Q: How do we find the point of intersection?

A: We can find the point of intersection by looking at the graph. The point of intersection is the point where the two equations intersect.

Q: What are some real-world applications of graphing?

A: Some real-world applications of graphing include:

• Science: Graphing is used in science to model real-world phenomena, such as the motion of objects and the behavior of populations.
• Engineering: Graphing is used in engineering to design and optimize systems, such as bridges and buildings.
• Economics: Graphing is used in economics to model economic systems and make predictions about future trends.

Q: What are some future research directions in graphing?

A: Some future research directions in graphing include:

• Developing new graphing algorithms: Developing new graphing algorithms that can handle complex equations and find their solutions more efficiently.
• Improving graphing software: Improving graphing software to make it more user-friendly and efficient.
• Applying graphing to real-world problems: Applying graphing to real-world problems, such as modeling population growth and predicting economic trends.

Conclusion

In conclusion, solving the equation $14 = \log_5(2x - 3)$ by graphing is a useful technique that can be applied to a variety of real-world problems. By using the change of base formula, graphing the system of equations, and finding the point of intersection, we can find the solution to the equation. Graphing has many real-world applications, including science, engineering, and economics. Future research directions in graphing include developing new graphing algorithms, improving graphing software, and applying graphing to real-world problems.

References

• Change of Base Formula: The change of base formula is a mathematical formula that allows us to rewrite a logarithmic equation in terms of a different base.
• Graphing Calculator: A graphing calculator is a tool that allows us to graph equations and find their solutions.
• Computer Program: A computer program is a tool that allows us to graph equations and find their solutions.

Glossary

• Change of Base Formula: A mathematical formula that allows us to rewrite a logarithmic equation in terms of a different base.
• Graphing Calculator: A tool that allows us to graph equations and find their solutions.
• Computer Program: A tool that allows us to graph equations and find their solutions.
• Point of Intersection: The solution to an equation, found by graphing the equation and looking for the point where the two equations intersect.