A Student Showed The Steps Below While Solving The Equation 14 = Log ⁡ 5 ( 2 X − 3 14=\log_5(2x-3 14 = Lo G 5 ​ ( 2 X − 3 ] By Graphing.Step 1: Write A System Of Equations: Y 1 = 14 Y_1 = 14 Y 1 ​ = 14 Y 2 = Log ⁡ 5 ( 2 X − 3 Y_2 = \log_5(2x-3 Y 2 ​ = Lo G 5 ​ ( 2 X − 3 ] Step 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. However, with the right approach and tools, it can be made more manageable. In this article, we will explore a student's approach to solving the equation $14=\log_5(2x-3)$ by graphing. We will break down the steps involved and provide a detailed explanation of each step.

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 visualize the problem and understand the relationship between the two variables, $x$ and $y$.

Step 2: Use the Change of Base Formula

The change of base formula is a mathematical formula that allows us to rewrite a logarithmic expression in terms of a different base. In this case, we want to rewrite the expression $\log_5(2x-3)$ in terms of a base that is more familiar to us.

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, and $c \neq 1$.

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

$$\log_5(2x-3) = \frac{\log(2x-3)}{\log(5)}$$

This allows us to rewrite the second equation in the system as:

$$y_2 = \frac{\log(2x-3)}{\log(5)}$$

Step 3: Graph the System of Equations

Now that we have rewritten the system of equations, we can graph the two equations on the same coordinate plane. The graph of the first equation, $y_1 = 14$, is a horizontal line at $y = 14$. The graph of the second equation, $y_2 = \frac{\log(2x-3)}{\log(5)}$, is a logarithmic curve.

By graphing the two equations, we can visualize the solution to the equation $14=\log_5(2x-3)$. The point of intersection between the two graphs represents the solution to the equation.

Step 4: Find the Point of Intersection

To find the point of intersection between the two graphs, we can set the two equations equal to each other and solve for $x$.

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

Solving for $x$, we get:

$$x = \frac{5^{14} + 3}{2}$$

This is the $x$-coordinate of the point of intersection. To find the $y$-coordinate, we can substitute this value of $x$ into either of the two equations.

Conclusion

In this article, we explored a student's approach to solving the equation $14=\log_5(2x-3)$ by graphing. We broke down the steps involved and provided a detailed explanation of each step. By using the change of base formula and graphing the system of equations, we were able to find the solution to the equation.

Key Takeaways

• The change of base formula allows us to rewrite a logarithmic expression in terms of a different base.
• Graphing the system of equations can help us visualize the solution to the equation.
• By finding the point of intersection between the two graphs, we can determine the solution to the equation.

Further Reading

For more information on solving equations involving logarithms, we recommend checking out the following resources:

• Khan Academy: Logarithms
• Mathway: Logarithmic Equations
• Wolfram Alpha: Logarithmic Equations

References

• [1] "Logarithms." Khan Academy, 2022.
• [2] "Logarithmic Equations." Mathway, 2022.
• [3] "Logarithmic Equations." Wolfram Alpha, 2022.

Glossary

• Change of Base Formula: A mathematical formula that allows us to rewrite a logarithmic expression in terms of a different base.
• Graphing: The process of visualizing a mathematical equation or function on a coordinate plane.
• Logarithmic Curve: A type of curve that is defined by a logarithmic equation.
• Point of Intersection: The point where two or more curves intersect on a coordinate plane.
  A Student's Approach to Solving the Equation $14=\log_5(2x-3)$ by Graphing: Q&A ====================================================================

Introduction

In our previous article, we explored a student's approach to solving the equation $14=\log_5(2x-3)$ by graphing. We broke down the steps involved and provided a detailed explanation of each step. In this article, we will answer some of the most 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 expression 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, and $c \neq 1$.

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 expression $\log_5(2x-3)$ in terms of a base that is more familiar to us. This allows us to graph the system of equations and find the point of intersection.

Q: How do we graph the system of equations?

A: To graph the system of equations, we need to graph the two equations on the same coordinate plane. The graph of the first equation, $y_1 = 14$, is a horizontal line at $y = 14$. The graph of the second equation, $y_2 = \frac{\log(2x-3)}{\log(5)}$, is a logarithmic curve.

Q: How do we find the point of intersection?

A: To find the point of intersection, we need to set the two equations equal to each other and solve for $x$. This gives us the $x$-coordinate of the point of intersection. To find the $y$-coordinate, we can substitute this value of $x$ into either of the two equations.

Q: What is the solution to the equation $14=\log_5(2x-3)$?

A: The solution to the equation $14=\log_5(2x-3)$ is given by:

$$x = \frac{5^{14} + 3}{2}$$

This is the $x$-coordinate of the point of intersection. To find the $y$-coordinate, we can substitute this value of $x$ into either of the two equations.

Q: Can we use other methods to solve the equation $14=\log_5(2x-3)$?

A: Yes, we can use other methods to solve the equation $14=\log_5(2x-3)$. Some of these methods include:

• Using the properties of logarithms to simplify the expression
• Using algebraic manipulations to isolate the variable
• Using numerical methods to approximate the solution

Q: What are some common mistakes to avoid when solving the equation $14=\log_5(2x-3)$?

A: Some common mistakes to avoid when solving the equation $14=\log_5(2x-3)$ include:

• Failing to use the change of base formula
• Graphing the system of equations incorrectly
• Failing to find the point of intersection
• Making algebraic errors when solving for $x$

Conclusion

In this article, we answered some of the most frequently asked questions about solving the equation $14=\log_5(2x-3)$ by graphing. We hope that this article has been helpful in clarifying some of the concepts involved in solving this equation.

Key Takeaways

• The change of base formula is a mathematical formula that allows us to rewrite a logarithmic expression in terms of a different base.
• Graphing the system of equations can help us visualize the solution to the equation.
• By finding the point of intersection between the two graphs, we can determine the solution to the equation.

Further Reading

For more information on solving equations involving logarithms, we recommend checking out the following resources:

• Khan Academy: Logarithms
• Mathway: Logarithmic Equations
• Wolfram Alpha: Logarithmic Equations

References

• [1] "Logarithms." Khan Academy, 2022.
• [2] "Logarithmic Equations." Mathway, 2022.
• [3] "Logarithmic Equations." Wolfram Alpha, 2022.

Glossary

• Change of Base Formula: A mathematical formula that allows us to rewrite a logarithmic expression in terms of a different base.
• Graphing: The process of visualizing a mathematical equation or function on a coordinate plane.
• Logarithmic Curve: A type of curve that is defined by a logarithmic equation.
• Point of Intersection: The point where two or more curves intersect on a coordinate plane.