1. **Problem:** Find the inverse of the logarithmic function $$y = \log_5(2x - 1) - 7$$. 2. **Recall the definition of inverse functions:** If $$y = f(x)$$, then the inverse function $$x = f^{-1}(y)$$ satisfies $$f(f^{-1}(y)) = y$$ and $$f^{-1}(f(x)) = x$$. 3. **Step to find the inverse:** Swap $$x$$ and $$y$$ and solve for $$y$$. Start with: $$y = \log_5(2x - 1) - 7$$ Swap $$x$$ and $$y$$: $$x = \log_5(2y - 1) - 7$$ 4. **Isolate the logarithm:** $$x + 7 = \log_5(2y - 1)$$ 5. **Rewrite the logarithmic equation in exponential form:** $$5^{x + 7} = 2y - 1$$ 6. **Solve for $$y$$:** $$2y = 5^{x + 7} + 1$$ $$y = \frac{5^{x + 7} + 1}{2}$$ 7. **Therefore, the inverse function is:** $$f^{-1}(x) = \frac{5^{x + 7} + 1}{2}$$ --- **Desmos graph for the original function:** $$y = \log_5(2x - 1) - 7$$ **Features:** intercepts and extrema are not typical for logarithmic functions shifted this way, but intercepts can be checked. --- "slug": "log inverse", "subject": "calculus", "desmos": { "latex": "y=\log_5(2x-1)-7", "features": { "intercepts": true, "extrema": true } }, "q_count": 2