Question

What is the vertical shift in f(x) = log base 3 (x^6/81) compared to its parent function

Answer 1

To analyze the vertical shift of the function \( f(x) = \log_{3}\left(\frac{x^6}{81}\right) \) compared to its parent function \( g(x) = \log_{3}(x) \), we can manipulate the expression for \( f(x) \).

We can rewrite \( f(x) \) using logarithmic properties:

\[
f(x) = \log_{3}\left(\frac{x^6}{81}\right) = \log_{3}(x^6) - \log_{3}(81)
\]

Using the property of logarithms that states \( \log_{b}(a^n) = n \log_{b}(a) \), we apply it to \( \log_{3}(x^6) \):

\[
f(x) = 6 \log_{3}(x) - \log_{3}(81)
\]

Next, we need to evaluate \( \log_{3}(81) \). Since \( 81 = 3^4 \), we have:

\[
\log_{3}(81) = 4
\]

Now substituting this back into our equation for \( f(x) \):

\[
f(x) = 6 \log_{3}(x) - 4
\]

From this expression, it’s clear that \( f(x) \) can be viewed as a transformed version of the parent function \( g(x) = \log_{3}(x) \). Specifically, \( f(x) \) involves:

1. A vertical stretch by a factor of 6 (due to the \( 6 \log_{3}(x) \)).
2. A vertical shift down by 4 units (due to the \( -4 \)).

Thus, the vertical shift of \( f(x) \) compared to its parent function \( g(x) \) is **downward by 4 units**.