To find the inverse function [tex]\( f^{-1}(x) \)[/tex] for the given function [tex]\( f(x) = 2^x \)[/tex], let's go through the process step-by-step.
1. Understand the Inverse Function:
- If [tex]\( f \)[/tex] maps an input [tex]\( x \)[/tex] to an output [tex]\( y \)[/tex], then the inverse function [tex]\( f^{-1} \)[/tex] should map [tex]\( y \)[/tex] back to [tex]\( x \)[/tex].
- Formally, if [tex]\( f(x) = y \)[/tex], then [tex]\( f^{-1}(y) = x \)[/tex].
2. Express the Original Function:
- We are given [tex]\( f(x) = 2^x \)[/tex].
3. Set Up the Equation for the Inverse:
- To find the inverse, we start by replacing [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]. So, [tex]\( y = 2^x \)[/tex].
4. Solve for [tex]\( x \)[/tex] in Terms of [tex]\( y \)[/tex]:
- We need to solve this equation for [tex]\( x \)[/tex]. Expressing [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] involves logarithms:
[tex]\[ y = 2^x \][/tex]
Taking the logarithm base 2 on both sides:
[tex]\[ \log_2(y) = \log_2(2^x) \][/tex]
Using the property of logarithms [tex]\( \log_b(a^c) = c\log_b(a) \)[/tex]:
[tex]\[ \log_2(y) = x \cdot \log_2(2) \][/tex]
5. Simplify:
- We know [tex]\( \log_2(2) = 1 \)[/tex], so:
[tex]\[ \log_2(y) = x \times 1 \][/tex]
[tex]\[ \log_2(y) = x \][/tex]
6. Express the Inverse Function:
- So, [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] is:
[tex]\[ x = \log_2(y) \][/tex]
- This gives us the inverse function:
[tex]\[ f^{-1}(x) = \log_2(x) \][/tex]
Therefore, the inverse function [tex]\( f^{-1}(x) \)[/tex] of [tex]\( f(x) = 2^x \)[/tex] is:
[tex]\[ f^{-1}(x) = \log_2(x) \][/tex]
Among the given choices, the correct one is:
[tex]\[ f^{-1}(x) = \log_2 x \][/tex]
Thus, the answer is [tex]\(\boxed{\log_2 x}\)[/tex].
