1. **Problem Statement:** Identify which logarithmic function matches the given graph description. 2. **Given Functions:** - $\log_{0.5} x$ - $\log_2 x$ - $\log |x|$ - $\ln x$ 3. **Graph Description Analysis:** - The graph passes through $(1,0)$, which is true for all logarithmic functions since $\log_b 1 = 0$ for any base $b$. - The curve starts near $y=4$ at $x=0$ and decreases rapidly. - The function decreases as $x$ increases, indicating a logarithm with base between 0 and 1. 4. **Key Logarithm Properties:** - For $b > 1$, $\log_b x$ is increasing. - For $0 < b < 1$, $\log_b x$ is decreasing. - $\log |x|$ is undefined for $x \leq 0$ and symmetric about the y-axis. - $\ln x$ is the natural logarithm with base $e > 1$, so it is increasing. 5. **Matching the Graph:** - The graph decreases as $x$ increases, so the base must be between 0 and 1. - Among the options, only $\log_{0.5} x$ has base $0.5$ which is between 0 and 1. 6. **Conclusion:** The graphed function is $\log_{0.5} x$. **Final answer:** $\boxed{\log_{0.5} x}$