How Would You Describe The Relationship Between The Real Zeros And $x$-intercepts Of The Function $y=\log _4(x-2$\]?A. When You Set The Function Equal To Zero, The Solution Is $x=6$; Therefore, The Graph Has An

Introduction

When dealing with logarithmic functions, it's essential to understand the relationship between the real zeros and x-intercepts. In this article, we'll explore the concept of real zeros and x-intercepts in the context of the function $y=\log _4(x-2)$. We'll delve into the properties of logarithmic functions, analyze the given function, and discuss how to find the real zeros and x-intercepts.

What are Real Zeros and x-Intercepts?

In mathematics, a real zero of a function is a value of the input variable (in this case, x) that makes the function equal to zero. On the other hand, an x-intercept is a point on the graph of a function where the function crosses the x-axis. In other words, an x-intercept is a point where the function has a value of zero.

Properties of Logarithmic Functions

Logarithmic functions have several properties that are essential to understand when working with them. One of the key properties is that the logarithm of a number is the exponent to which a base must be raised to produce that number. In the case of the function $y=\log _4(x-2)$, the base is 4, and the argument is $(x-2)$.

Analyzing the Given Function

The given function is $y=\log _4(x-2)$. To find the real zeros and x-intercepts of this function, we need to set the function equal to zero and solve for x.

Finding the Real Zeros

To find the real zeros of the function, we set $y=\log _4(x-2)$ equal to zero and solve for x.

$$\log _4(x-2) = 0$$

Using the definition of a logarithm, we can rewrite this equation as:

$$4^0 = x-2$$

Simplifying the equation, we get:

$$1 = x-2$$

Adding 2 to both sides, we get:

$$x = 3$$

Therefore, the real zero of the function is $x=3$.

Finding the x-Intercepts

To find the x-intercepts of the function, we need to find the points where the function crosses the x-axis. In other words, we need to find the values of x that make the function equal to zero.

As we've already seen, the real zero of the function is $x=3$. This means that the graph of the function crosses the x-axis at the point $(3,0)$.

Conclusion

In conclusion, the real zero of the function $y=\log _4(x-2)$ is $x=3$, and the x-intercept is the point $(3,0)$. Understanding the relationship between real zeros and x-intercepts is essential when working with logarithmic functions.

Real Zeros and x-Intercepts: A Comparison

In this section, we'll compare the real zeros and x-intercepts of the function $y=\log _4(x-2)$.

As we've seen, the real zero of the function is $x=3$, and the x-intercept is the point $(3,0)$. This means that the real zero and x-intercept are the same point.

Why is this the case?

The reason why the real zero and x-intercept are the same point is because the function $y=\log _4(x-2)$ is a decreasing function. This means that as x increases, the value of the function decreases.

What does this mean for the graph of the function?

The fact that the function is decreasing means that the graph of the function will be a decreasing curve. This means that as x increases, the value of the function will decrease, and the graph will approach the x-axis.

What are the implications of this for the real zeros and x-intercepts?

The fact that the function is decreasing means that there will be only one real zero and one x-intercept. This is because the function will only cross the x-axis once, at the point where the real zero and x-intercept are located.

Conclusion

In conclusion, the real zero and x-intercept of the function $y=\log _4(x-2)$ are the same point, $(3,0)$. This is because the function is a decreasing function, and the graph of the function will approach the x-axis as x increases.

Real Zeros and x-Intercepts: A Graphical Approach

In this section, we'll use a graphical approach to understand the relationship between real zeros and x-intercepts.

Graphing the Function

To graph the function $y=\log _4(x-2)$, we can use a graphing calculator or software.

What does the graph look like?

The graph of the function will be a decreasing curve that approaches the x-axis as x increases.

Where is the real zero located?

The real zero of the function is located at the point where the graph crosses the x-axis. In this case, the real zero is located at the point $(3,0)$.

What does this mean for the x-intercept?

The x-intercept is the point where the graph crosses the x-axis. In this case, the x-intercept is also located at the point $(3,0)$.

Conclusion

In conclusion, the graph of the function $y=\log _4(x-2)$ shows that the real zero and x-intercept are the same point, $(3,0)$. This is because the function is a decreasing function, and the graph of the function will approach the x-axis as x increases.

Real Zeros and x-Intercepts: A Summary

In this article, we've explored the relationship between real zeros and x-intercepts in the context of the function $y=\log _4(x-2)$. We've seen that the real zero and x-intercept are the same point, $(3,0)$, and that this is due to the fact that the function is a decreasing function.

Key Takeaways

• The real zero of the function $y=\log _4(x-2)$ is $x=3$.
• The x-intercept of the function is the point $(3,0)$.
• The function is a decreasing function, and the graph of the function will approach the x-axis as x increases.
• The real zero and x-intercept are the same point due to the fact that the function is a decreasing function.

Conclusion

In conclusion, understanding the relationship between real zeros and x-intercepts is essential when working with logarithmic functions. By analyzing the properties of logarithmic functions and using graphical approaches, we can gain a deeper understanding of this relationship.

Introduction

In our previous article, we explored the relationship between real zeros and x-intercepts in the context of the function $y=\log _4(x-2)$. In this article, we'll answer some of the most frequently asked questions about real zeros and x-intercepts.

Q: What is the difference between a real zero and an x-intercept?

A: A real zero is a value of the input variable (in this case, x) that makes the function equal to zero. An x-intercept, on the other hand, is a point on the graph of a function where the function crosses the x-axis.

Q: How do you find the real zeros of a function?

A: To find the real zeros of a function, you need to set the function equal to zero and solve for x. This will give you the values of x that make the function equal to zero.

Q: How do you find the x-intercepts of a function?

A: To find the x-intercepts of a function, you need to find the points where the function crosses the x-axis. This can be done by setting the function equal to zero and solving for x.

Q: What is the relationship between real zeros and x-intercepts?

A: The real zeros and x-intercepts of a function are the same point. This is because the function crosses the x-axis at the point where the real zero is located.

Q: Why is this the case?

A: This is the case because the function is a decreasing function. This means that as x increases, the value of the function decreases, and the graph of the function will approach the x-axis.

Q: What are the implications of this for the real zeros and x-intercepts?

A: The fact that the function is decreasing means that there will be only one real zero and one x-intercept. This is because the function will only cross the x-axis once, at the point where the real zero and x-intercept are located.

Q: Can you give an example of a function where the real zero and x-intercept are not the same point?

A: Yes, consider the function $y=x^2-4$. The real zeros of this function are $x=2$ and $x=-2$, and the x-intercepts are the points $(2,0)$ and $(-2,0)$. In this case, the real zeros and x-intercepts are not the same point.

Q: Why is this the case?

A: This is the case because the function $y=x^2-4$ is an increasing function. This means that as x increases, the value of the function increases, and the graph of the function will move away from the x-axis.

Q: What are the implications of this for the real zeros and x-intercepts?

A: The fact that the function is increasing means that there will be two real zeros and two x-intercepts. This is because the function will cross the x-axis twice, at the points where the real zeros are located.

Q: Can you give an example of a function where the real zero and x-intercept are the same point, but the function is not a decreasing function?

A: Yes, consider the function $y=\log _4(x-2)$. The real zero of this function is $x=3$, and the x-intercept is the point $(3,0)$. In this case, the real zero and x-intercept are the same point, but the function is not a decreasing function.

Q: Why is this the case?

A: This is the case because the function $y=\log _4(x-2)$ is a logarithmic function with a base of 4. This means that the function will approach the x-axis as x increases, but it will not decrease.

Q: What are the implications of this for the real zeros and x-intercepts?

A: The fact that the function is a logarithmic function means that there will be only one real zero and one x-intercept. This is because the function will only cross the x-axis once, at the point where the real zero and x-intercept are located.

Conclusion

In conclusion, understanding the relationship between real zeros and x-intercepts is essential when working with functions. By analyzing the properties of functions and using graphical approaches, we can gain a deeper understanding of this relationship.