Let’s analyze the key points and characteristics of the function $y = f(x) = \log_{1/2} x$.
The equation $y = \log_{1/2} x$ is equivalent to the exponential equation $x = \left(\frac{1}{2}\right)^y$. We can use this to easily find corresponding $(x, y)$ values:
| x Value | Calculation | y Value | Point (x,y) |
| 1 | $y = \log_{1/2} 1 = 0$ | 0 | (1, 0) |
| 2 | $y = \log_{1/2} 2 = \log_{1/2} \left(\frac{1}{2}\right)^{-1} = -1$ | -1 | (2, -1) |
| 4 | $y = \log_{1/2} 4 = -2$ | -2 | (4, -2) |
| $\mathbf{\frac{1}{2}}$ | $y = \log_{1/2} \left(\frac{1}{2}\right) = 1$ | 1 | ($\frac{1}{2}$, 1) |
| $\mathbf{\frac{1}{4}}$ | $y = \log_{1/2} \left(\frac{1}{4}\right) = 2$ | 2 | ($\frac{1}{4}$, 2) |
Behavior: The graph is a decreasing function. As $x$ increases, the value of $y$ decreases.
X-Intercept: The graph crosses the x-axis at (1, 0). This is true for all basic logarithmic functions, as $\log_b 1 = 0$.
Asymptote: The graph has a vertical asymptote at $x = 0$ (the $Y$-axis). The curve gets closer and closer to the $Y$-axis but never touches or crosses it.
Domain: Since the argument of the logarithm must be positive, the domain is $x > 0$.
Range: The range includes all real numbers, $(-\infty, \infty)$.
This function is a reflection of $y = \log_2 x$ across the x-axis.