A Student Solved The Equation Below By Graphing.$\log _6(x-1) = \log _2(2x+2$\]Which Statement About The Graph Is True?A. The Curves Do Not Intersect.B. The Curves Intersect At One Point.C. The Curves Intersect At Two Points.D. The Curves

A Student's Graphing Conundrum: Unraveling the Mystery of Logarithmic Equations

In the realm of mathematics, logarithmic equations can be a daunting task for students to tackle. However, with the right approach and tools, these equations can be solved with ease. In this article, we will delve into a specific logarithmic equation that was solved by a student using graphing techniques. We will examine the equation, understand the graphing process, and determine which statement about the graph is true.

The equation in question is $\log _6(x-1) = \log _2(2x+2)$. This equation involves logarithms with different bases, which can make it challenging to solve. However, by using graphing techniques, the student was able to find a solution.

To graph the equation, we need to first understand the properties of logarithmic functions. The graph of a logarithmic function with base $b$ is a curve that increases as the input value increases. The curve has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$.

In this case, we have two logarithmic functions with different bases: $\log _6(x-1)$ and $\log _2(2x+2)$. To graph these functions, we can use a graphing calculator or software.

Graphing the Functions

When we graph the functions $\log _6(x-1)$ and $\log _2(2x+2)$, we get two curves that intersect at a single point. This is because the two functions are equal at this point, and the graph of the equation is the set of all points that satisfy the equation.

Now that we have graphed the equation, let's analyze the graph to determine which statement about the graph is true.

• A. The curves do not intersect. This statement is false, as we can see from the graph that the two curves intersect at a single point.
• B. The curves intersect at one point. This statement is true, as we can see from the graph that the two curves intersect at a single point.
• C. The curves intersect at two points. This statement is false, as we can see from the graph that the two curves intersect at only one point.
• D. The curves do not intersect at all. This statement is false, as we can see from the graph that the two curves intersect at a single point.

In conclusion, the student who solved the equation $\log _6(x-1) = \log _2(2x+2)$ using graphing techniques was able to find a solution by analyzing the graph of the equation. The graph shows that the two curves intersect at a single point, which means that the equation has a single solution.

• Logarithmic equations can be solved using graphing techniques.
• The graph of a logarithmic equation is a curve that increases as the input value increases.
• The graph of a logarithmic equation has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$.
• The graph of the equation $\log _6(x-1) = \log _2(2x+2)$ shows that the two curves intersect at a single point.

In this article, we have seen how a student solved a logarithmic equation using graphing techniques. We have analyzed the graph of the equation and determined which statement about the graph is true. By understanding the properties of logarithmic functions and using graphing techniques, students can solve logarithmic equations with ease.
A Student's Graphing Conundrum: Unraveling the Mystery of Logarithmic Equations

In our previous article, we explored a logarithmic equation that was solved by a student using graphing techniques. We analyzed the graph of the equation and determined which statement about the graph is true. In this article, we will answer some frequently asked questions about graphing logarithmic equations.

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

A: A logarithmic function is the inverse of an exponential function. While an exponential function grows rapidly as the input value increases, a logarithmic function grows slowly as the input value increases. In other words, an exponential function is the opposite of a logarithmic function.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you can use a graphing calculator or software. You can also use a table of values to plot the function. The graph of a logarithmic function is a curve that increases as the input value increases. The curve has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$.

Q: What is the domain of a logarithmic function?

A: The domain of a logarithmic function is all real numbers greater than 0. This is because the logarithm of a non-positive number is undefined.

Q: How do I find the intersection point of two logarithmic functions?

A: To find the intersection point of two logarithmic functions, you can set the two functions equal to each other and solve for the input value. This will give you the x-coordinate of the intersection point. You can then substitute this value into one of the functions to find the y-coordinate of the intersection point.

Q: Can I use graphing techniques to solve logarithmic equations with different bases?

A: Yes, you can use graphing techniques to solve logarithmic equations with different bases. However, you need to make sure that the bases are not too large or too small, as this can affect the accuracy of the graph.

Q: How do I determine which statement about the graph is true?

A: To determine which statement about the graph is true, you need to analyze the graph carefully. Look for the intersection point of the two curves, and determine whether the curves intersect at one point, two points, or not at all.

Q: Can I use graphing techniques to solve logarithmic equations with multiple variables?

A: Yes, you can use graphing techniques to solve logarithmic equations with multiple variables. However, you need to make sure that the equation is simplified and that the variables are isolated.

Q: How do I check my work when using graphing techniques to solve logarithmic equations?

A: To check your work when using graphing techniques to solve logarithmic equations, you need to verify that the intersection point of the two curves satisfies the equation. You can do this by substituting the x-coordinate and y-coordinate of the intersection point into the equation and checking if it is true.

In conclusion, graphing logarithmic equations can be a powerful tool for solving these types of equations. By understanding the properties of logarithmic functions and using graphing techniques, students can solve logarithmic equations with ease. We hope that this article has answered some of your frequently asked questions about graphing logarithmic equations.

• Logarithmic equations can be solved using graphing techniques.
• The graph of a logarithmic equation is a curve that increases as the input value increases.
• The graph of a logarithmic equation has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$.
• The graph of the equation $\log _6(x-1) = \log _2(2x+2)$ shows that the two curves intersect at a single point.

In this article, we have seen how graphing techniques can be used to solve logarithmic equations. We have answered some frequently asked questions about graphing logarithmic equations and provided some key takeaways. By understanding the properties of logarithmic functions and using graphing techniques, students can solve logarithmic equations with ease.