What Is The Vertical Shift In $f(x)=\log _3\left(\frac{x^6}{81}\right)$ Compared To Its Parent Function?A. 4 Units Up B. 4 Units Down C. 6 Units Up D. 6 Units Down

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Introduction

In mathematics, a vertical shift in a function refers to a change in the position of the graph of the function along the y-axis. This type of shift can be either upward or downward, depending on the direction of the shift. In this article, we will explore the concept of vertical shifts in logarithmic functions, specifically in the function $f(x)=\log _3\left(\frac{x^6}{81}\right)$. We will analyze the vertical shift of this function compared to its parent function and determine the correct answer from the given options.

Parent Function

The parent function of the given function is $f(x)=\log _3x$. This is a basic logarithmic function with base 3. The graph of this function is a logarithmic curve that increases as x increases.

Given Function

The given function is $f(x)=\log _3\left(\frac{x^6}{81}\right)$. To understand the vertical shift of this function, we need to analyze the expression inside the logarithm. The expression is $\frac{x^6}{81}$.

Simplifying the Expression

We can simplify the expression by rewriting 81 as $3^4$. This gives us:

\frac{x^6}{81} = \frac{x^6}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^4} = \frac{x^6}{3^4} \cdot \frac{3^4}{3^<br/> **Understanding Vertical Shifts in Logarithmic Functions** =========================================================== **Q&A: Vertical Shifts in Logarithmic Functions** ---------------------------------------------- **Q: What is a vertical shift in a function?** ------------------------------------------ A: A vertical shift in a function refers to a change in the position of the graph of the function along the y-axis. This type of shift can be either upward or downward, depending on the direction of the shift. **Q: How do you determine the vertical shift of a logarithmic function?** ---------------------------------------------------------------- A: To determine the vertical shift of a logarithmic function, you need to analyze the expression inside the logarithm. If the expression is in the form $\frac{x^a}{b^c}$, where a and c are positive integers, then the vertical shift is determined by the value of c. **Q: What is the vertical shift of the function $f(x)=\log _3\left(\frac{x^6}{81}\right)$ compared to its parent function?** --------------------------------------------------------- A: To determine the vertical shift of the function $f(x)=\log _3\left(\frac{x^6}{81}\right)$, we need to analyze the expression inside the logarithm. The expression is $\frac{x^6}{81}$. We can simplify the expression by rewriting 81 as $3^4$. This gives us: $\frac{x^6}{81} = \frac{x^6}{3^4}

Since the expression is in the form $\frac{xa}{bc}$, where a = 6 and c = 4, the vertical shift is determined by the value of c.

The vertical shift is 4 units down.

Q: What is the parent function of the given function?

A: The parent function of the given function is $f(x)=\log _3x$. This is a basic logarithmic function with base 3.

Q: How do you simplify the expression inside the logarithm?

A: To simplify the expression inside the logarithm, you can rewrite the number 81 as $3^4$. This gives us:

x681=x634\frac{x^6}{81} = \frac{x^6}{3^4}

Q: What is the effect of the vertical shift on the graph of the function?

A: The vertical shift affects the position of the graph of the function along the y-axis. If the vertical shift is upward, the graph will be shifted up. If the vertical shift is downward, the graph will be shifted down.

Q: How do you determine the direction of the vertical shift?

A: To determine the direction of the vertical shift, you need to analyze the expression inside the logarithm. If the expression is in the form $\frac{xa}{bc}$, where a and c are positive integers, then the vertical shift is determined by the value of c.

If c is greater than 1, the vertical shift is upward. If c is less than 1, the vertical shift is downward.

Q: What is the correct answer from the given options?

A: The correct answer is B. 4 units down.

Conclusion

In this article, we have discussed the concept of vertical shifts in logarithmic functions. We have analyzed the vertical shift of the function $f(x)=\log _3\left(\frac{x^6}{81}\right)$ compared to its parent function and determined the correct answer from the given options. We have also provided a Q&A section to help readers understand the concept of vertical shifts in logarithmic functions.