91Ó°ÊÓ

When we're talking about a decreasing function, we're usually referring to the way the values of the function reduce as you move along the x-axis. For the logarithmic function given by \(f(x) = \log_{1/2} x\), the base of the logarithm is \(\frac{1}{2}\). This fractional base (between 0 and 1) is the reason the function decreases. Imagine you're taking smaller and smaller steps downward as you move right; that's essentially what's happening here. For each increase in x, the output \(f(x)\) (or y-value) gets smaller. The graph starts high on the left and slopes downward to the right, which visually represents a decreasing function. This property of decreasing is important in many applications, especially when dealing with phenomena that reduce over time, like radioactive decay or cooling processes.

A vertical asymptote is like an invisible barrier that a function's graph approaches but never quite touches or crosses. For the logarithmic function \(f(x) = \log_{1/2} x\), this barrier occurs at the line \(x = 0\). Here's why: for any logarithmic function, the input values (x-values) must be greater than zero. If you try to plug \(x = 0\) into the logarithm, the expression becomes undefined—like trying to divide by zero.As you move closer to \(x = 0\) from the positive side, the output of the function dives sharply down toward negative infinity. You can think of \(x = 0\) as the edge of a cliff; you can lean closer to the edge, but you'll never step over it. So, the vertical asymptote represents where the function 'breaks' or is undefined.

Understanding the x-intercept is critical when graphing any function, including logarithmic ones like \(f(x) = \log_{1/2} x\). The x-intercept is the point where the function crosses the x-axis, meaning the y-value (or \(f(x)\)) is zero. To find this spot for our function, we set \(\log_{1/2} x = 0\). This can be re-expressed using exponents: \(\left(\frac{1}{2}\right)^0 = x\). Solving for x gives us \(x = 1\), so the x-intercept is at the point \((1, 0)\).This point is where the function has a value of zero, and as you move to the left or right from this spot, the function's value changes, providing a useful anchor for graphing the function's behavior.

The base of a logarithm significantly influences the function's behavior. In the case of the function \(f(x) = \log_{1/2} x\), the base is \(\frac{1}{2}\). This base decides two crucial properties:

• Since \(\frac{1}{2}\) is between 0 and 1, the function is decreasing, as each additional unit increase in x results in a decrease in the function's output.
• The base also dictates the rate at which these changes occur. Being smaller than 1 means any power of \(\frac{1}{2}\) will be less than 1, which compresses the scale of the function's graph horizontally.

Changing the base of the logarithm shifts the entire nature of the function. If the base were greater than 1, we'd see an increasing behavior instead, and the steepness and direction of the graph's curve would change accordingly. Thus, grasping the base is essential for predicting and understanding the logarithmic function's characteristics.