What is the vertical shift in f(x)=log3(x6/81)

Question

as compared to its parent function?(1 point)
Responses

6 units up
6 units up

4 units down
4 units down

6 units down
6 units down

4 units up

Answer 1

To determine the vertical shift of the function \( f(x) = \log_3\left(\frac{x^6}{81}\right) \) compared to its parent function \( f(x) = \log_3(x) \), we can rewrite the given function in a more manageable form.

We begin by using logarithmic properties:

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

Now, we can simplify this further:

1. Using the power rule of logarithms, we get: \[ \log_3(x^6) = 6 \log_3(x) \]

2. The term \( \log_3(81) \) can be computed since \( 81 = 3^4 \): \[ \log_3(81) = \log_3(3^4) = 4 \]

Putting this all together, we have:

\[ f(x) = 6 \log_3(x) - 4 \]

In this form, we see that the function has a multiplier of 6 (which is a vertical stretch) and a constant \( -4 \) which indicates a vertical shift.

Since the constant \( -4 \) indicates a downward shift, we can conclude that the vertical shift compared to the parent function \( f(x) = \log_3(x) \) is:

4 units down.

So the correct response would be:

4 units down.