To determine which function is the inverse of [tex]\( y = 3^x \)[/tex], follow these steps:
1. Understand the Definition of an Inverse Function:
The inverse function of [tex]\( y = 3^x \)[/tex] should swap the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] and solve for the new [tex]\( y \)[/tex]. That is, we start with [tex]\( y = 3^x \)[/tex] and find [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
y = 3^x
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
To find [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex], take the logarithm base 3 of both sides:
[tex]\[
\log_3 y = \log_3 (3^x)
\][/tex]
3. Apply Logarithm Properties:
Using the property of logarithms that [tex]\( \log_b (b^k) = k \)[/tex], we get:
[tex]\[
\log_3 y = x
\][/tex]
Therefore, the inverse function is:
[tex]\[
x = \log_3 y
\][/tex]
4. Rewrite the Inverse Function:
Labeling the inverse function as [tex]\( y \)[/tex] again, the inverse relation can be written as:
[tex]\[
y = \log_3 x
\][/tex]
5. Match the Given Choices:
Now, let's compare this inverse function [tex]\( y = \log_3 x \)[/tex] with the provided choices:
- [tex]\( y = \frac{1}{3^x} \)[/tex]
- [tex]\( y = \log_3 x \)[/tex]
- [tex]\( y = \left(\frac{1}{3}\right)^x \)[/tex]
- [tex]\( y = \log_{\frac{1}{3}} x \)[/tex]
6. Identify the Correct Answer:
From the choices, the correct inverse function matches [tex]\( y = \log_3 x \)[/tex].
Thus, the inverse of [tex]\( y = 3^x \)[/tex] is:
[tex]\[
y = \log_3 x
\][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{y = \log_3 x} \][/tex]
