To determine the inverse function of [tex]\( F(x) = b^x \)[/tex], we need to find a function [tex]\( F^{-1}(y) \)[/tex] such that [tex]\( F(F^{-1}(y)) = y \)[/tex].
1. Let's start with the given function:
[tex]\[ F(x) = b^x \][/tex]
2. To find the inverse function [tex]\( F^{-1}(y) \)[/tex], we need to switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex], and then solve for [tex]\( x \)[/tex]:
[tex]\[ y = b^x \][/tex]
3. To isolate [tex]\( x \)[/tex], we'll take the logarithm with base [tex]\( b \)[/tex] on both sides of the equation:
[tex]\[ \log_b y = \log_b (b^x) \][/tex]
4. Using the logarithm property [tex]\( \log_b (b^x) = x \)[/tex], we get:
[tex]\[ x = \log_b y \][/tex]
So, the inverse function [tex]\( F^{-1}(y) \)[/tex] is:
[tex]\[ F^{-1}(y) = \log_b y \][/tex]
Therefore, the correct answer is:
[tex]\[ \text{D. } F^{-1}(y) = \log_b y \][/tex]
