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

Understanding Logarithmic Functions

A logarithmic function is a mathematical function that is the inverse of an exponential function. It is a function that takes a number as input and returns a value that represents the power to which a base number must be raised to produce the input number. The general form of a logarithmic function is:

$$f(x) = \log_b(x)$$

where $b$ is the base of the logarithm and $x$ is the input number.

Vertical Shift in Logarithmic Functions

The vertical shift in a logarithmic function refers to the change in the output value of the function when the input value is changed. In other words, it refers to the change in the y-value of the function when the x-value is changed.

The vertical shift in a logarithmic function can be determined by comparing the function to its parent function. The parent function of a logarithmic function is the function without any vertical shifts or horizontal shifts.

Vertical Shift in the Given Function

The given function is:

$$f(x) = \log_3\left(\frac{x^6}{81}\right)$$

To determine the vertical shift in this function, we need to compare it to its parent function. The parent function of this function is:

$$f(x) = \log_3(x)$$

The given function can be rewritten as:

$$f(x) = \log_3\left(\frac{x^6}{81}\right) = \log_3(x^6) - \log_3(81)$$

Using the property of logarithms that $\log_b(x^a) = a\log_b(x)$, we can rewrite the function as:

$$f(x) = 6\log_3(x) - \log_3(81)$$

Since $\log_3(81) = 4$, we can rewrite the function as:

$$f(x) = 6\log_3(x) - 4$$

Comparing this function to its parent function, we can see that the given function has a vertical shift of $-4$ units.

Conclusion

In conclusion, the vertical shift in the given function $f(x) = \log_3\left(\frac{x^6}{81}\right)$ as compared to its parent function is $-4$ units. This means that the given function is shifted $4$ units down from its parent function.

Answer

The correct answer is:

B. 4 units down

Key Takeaways

• The vertical shift in a logarithmic function refers to the change in the output value of the function when the input value is changed.
• The vertical shift in a logarithmic function can be determined by comparing the function to its parent function.
• The parent function of a logarithmic function is the function without any vertical shifts or horizontal shifts.
• The given function $f(x) = \log_3\left(\frac{x^6}{81}\right)$ has a vertical shift of $-4$ units as compared to its parent function.

Practice Problems

1. Find the vertical shift in the function $f(x) = \log_2\left(\frac{x^3}{8}\right)$ as compared to its parent function.
2. Find the vertical shift in the function $f(x) = \log_5\left(\frac{x^2}{25}\right)$ as compared to its parent function.

Solutions

1. The vertical shift in the function $f(x) = \log_2\left(\frac{x^3}{8}\right)$ as compared to its parent function is $-3$ units.
2. The vertical shift in the function $f(x) = \log_5\left(\frac{x^2}{25}\right)$ as compared to its parent function is $-2$ units.
   General Form of a Logarithmic Function Quick Check =====================================================

Q&A: General Form of a Logarithmic Function

Q: What is the general form of a logarithmic function?

A: The general form of a logarithmic function is:

$$f(x) = \log_b(x)$$

where $b$ is the base of the logarithm and $x$ is the input number.

Q: What is the parent function of a logarithmic function?

A: The parent function of a logarithmic function is the function without any vertical shifts or horizontal shifts. It is the function that has the base as the only parameter.

Q: How do you determine the vertical shift in a logarithmic function?

A: To determine the vertical shift in a logarithmic function, you need to compare the function to its parent function. The vertical shift is the difference between the output value of the function and the output value of the parent function.

Q: What is the vertical shift in the function $f(x) = \log_3\left(\frac{x^6}{81}\right)$ as compared to its parent function?

A: The vertical shift in the function $f(x) = \log_3\left(\frac{x^6}{81}\right)$ as compared to its parent function is $-4$ units.

Q: How do you rewrite a logarithmic function in a different form?

A: You can rewrite a logarithmic function in a different form by using the properties of logarithms. For example, you can use the property $\log_b(x^a) = a\log_b(x)$ to rewrite the function.

Q: What is the property of logarithms that states $\log_b(x^a) = a\log_b(x)$?

A: This property is known as the power rule of logarithms.

Q: How do you use the power rule of logarithms to rewrite a logarithmic function?

A: To use the power rule of logarithms, you need to identify the exponent of the input number and multiply it by the logarithm of the base.

Q: What is the vertical shift in the function $f(x) = \log_2\left(\frac{x^3}{8}\right)$ as compared to its parent function?

A: The vertical shift in the function $f(x) = \log_2\left(\frac{x^3}{8}\right)$ as compared to its parent function is $-3$ units.

Q: What is the vertical shift in the function $f(x) = \log_5\left(\frac{x^2}{25}\right)$ as compared to its parent function?

A: The vertical shift in the function $f(x) = \log_5\left(\frac{x^2}{25}\right)$ as compared to its parent function is $-2$ units.

Q: How do you determine the horizontal shift in a logarithmic function?

A: To determine the horizontal shift in a logarithmic function, you need to compare the function to its parent function. The horizontal shift is the difference between the input value of the function and the input value of the parent function.

Q: What is the horizontal shift in the function $f(x) = \log_3\left(\frac{x^6}{81}\right)$ as compared to its parent function?

A: The horizontal shift in the function $f(x) = \log_3\left(\frac{x^6}{81}\right)$ as compared to its parent function is $0$ units.

Q: How do you use the properties of logarithms to simplify a logarithmic expression?

A: You can use the properties of logarithms to simplify a logarithmic expression by combining the logarithms of the same base.

Q: What is the property of logarithms that states $\log_b(x) + \log_b(y) = \log_b(xy)$?

A: This property is known as the product rule of logarithms.

Q: How do you use the product rule of logarithms to simplify a logarithmic expression?

A: To use the product rule of logarithms, you need to combine the logarithms of the same base by multiplying the input numbers.

Q: What is the property of logarithms that states $\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)$?

A: This property is known as the quotient rule of logarithms.

Q: How do you use the quotient rule of logarithms to simplify a logarithmic expression?

A: To use the quotient rule of logarithms, you need to combine the logarithms of the same base by dividing the input numbers.