Logarithm Inverse A74031

1. **Problem Statement:** We are given the function $$f(x) = \log_2 x$$ and asked to: a) Sketch its graph. b) Find its inverse function. c) Sketch the inverse on the same diagram. d) Complete a table comparing properties of $$f(x)$$ and its inverse. e) Conclude about logarithmic functions and their inverses. 2. **Recall the definition and properties:** - The logarithmic function $$f(x) = \log_2 x$$ is defined for $$x > 0$$. - Its range is all real numbers $$(-\infty, \infty)$$. - The inverse function of a logarithm is an exponential function. 3. **Find the inverse function:** Start with $$y = \log_2 x$$. Rewrite in exponential form: $$x = 2^y$$. Swap $$x$$ and $$y$$ to find the inverse: $$y = 2^x$$. So the inverse function is: $$f^{-1}(x) = 2^x$$. 4. **Sketching the graphs:** - The graph of $$f(x) = \log_2 x$$ is an increasing curve passing through $$(1,0)$$, approaching the y-axis (vertical asymptote at $$x=0$$) but never touching it. - The graph of $$f^{-1}(x) = 2^x$$ is an increasing exponential curve passing through $$(0,1)$$, approaching the x-axis (horizontal asymptote at $$y=0$$) but never touching it. - Both graphs are symmetric about the line $$y = x$$. 5. **Complete the table:** | Property | $$f(x) = \log_2 x$$ | $$f^{-1}(x) = 2^x$$ | |----------------|----------------------------|---------------------------| | Domain | $$x > 0$$ | $$x \in (-\infty, \infty)$$ | | Range | $$y \in (-\infty, \infty)$$ | $$y > 0$$ | | x-intercept | $$(1,0)$$ | None | | y-intercept | None | $$(0,1)$$ | | Asymptotes | Vertical at $$x=0$$ | Horizontal at $$y=0$$ | 6. **Conclusion:** - Logarithmic functions and their inverses (exponential functions) are reflections of each other across the line $$y = x$$. - The domain of one is the range of the other and vice versa. - The asymptotes of one correspond to the intercepts or limits of the other. **Final answer:** $$f^{-1}(x) = 2^x$$