1. **State the problem:** We need to find the equation of the logarithmic function $f(x)$ given its graph. 2. **Identify key features:** The graph has a vertical asymptote at $x=4$, which means the logarithmic function is shifted horizontally by 4 units. 3. **General form of translated logarithmic function:** $$f(x) = a \log_b(x - h) + k$$ where $h$ is the horizontal shift, $k$ is the vertical shift, $a$ is a vertical stretch/compression, and $b$ is the base of the logarithm. 4. **From the asymptote:** Since the vertical asymptote is at $x=4$, we have $h=4$. 5. **Use points to find $a$, $b$, and $k$:** The graph passes near points $(5,4)$ and $(10,5)$. 6. **Set up equations:** $$f(5) = a \log_b(5 - 4) + k = a \log_b(1) + k = k = 4$$ Since $\log_b(1) = 0$, this gives $k=4$. 7. **Use second point:** $$f(10) = a \log_b(10 - 4) + 4 = a \log_b(6) + 4 = 5$$ 8. **Solve for $a \log_b(6)$:** $$a \log_b(6) = 5 - 4 = 1$$ 9. **Assuming base 10 logarithm (common log):** $$\log_{10}(6) \approx 0.77815$$ 10. **Solve for $a$:** $$a = \frac{1}{0.77815} \approx 1.285$$ 11. **Final function:** $$f(x) = 1.285 \log_{10}(x - 4) + 4$$ This function matches the given graph with vertical asymptote at $x=4$ and passing near the given points.