Problem 43 - Intermediate Algebra

Question

For Problems \(41-46\), follow the suggested procedure to graph logarithmic functions. Graph \(y=\log _{\frac{1}{2}} x\) by graphing \(\left(\frac{1}{2}\right)^{y}=x\).

Step-by-Step Solution

Answer

Graph \( y = \log_{\frac{1}{2}} x \) using its points derived from \( x = (\frac{1}{2})^y \).

Step 1: Understand the Relationship between the Two Functions

The logarithmic function \(y = \log_{\frac{1}{2}} x\) can be rewritten in exponential form as \(x = (\frac{1}{2})^y\). This is the inverse of the given logarithmic function. Understanding that one is the inverse of the other helps in graphing the function.

Step 2: Identify Key Points for the Exponential Function

To graph \( x = (\frac{1}{2})^y \), choose several values for \(y\) and calculate corresponding \(x\) values. For example: - When \( y = -2 \), \( x = (\frac{1}{2})^{-2} = 4 \).- When \( y = -1 \), \( x = (\frac{1}{2})^{-1} = 2 \).- When \( y = 0 \), \( x = (\frac{1}{2})^{0} = 1 \).- When \( y = 1 \), \( x = (\frac{1}{2})^{1} = \frac{1}{2} \).- When \( y = 2 \), \( x = (\frac{1}{2})^{2} = \frac{1}{4} \).

Step 3: Plot the Exponential Graph

Using the key points identified, plot the reciprocal graph \(x = (\frac{1}{2})^y\) on the coordinate plane. This graph will be helpful since it is a reflection across the line \(y=x\) of the logarithmic function. Represent points such as (4, -2), (2, -1), (1, 0), (\(\frac{1}{2}\), 1), (\(\frac{1}{4}\), 2) on the plane.

Step 4: Convert the Exponential Graph to the Logarithmic Graph

The graph of \(y = \log_{\frac{1}{2}} x\) can be obtained by swapping the \(x\) and \(y\) coordinates from the points of the exponential graph. This means the logarithmic graph will pass through the points (-2, 4), (-1, 2), (0, 1), (1, \(\frac{1}{2}\)), and (2, \(\frac{1}{4}\)).

Step 5: Plot the Logarithmic Graph

Using the information and points from the previous step, plot the graph of \(y = \log_{\frac{1}{2}} x\). This graph decreases as \(x\) increases, indicating it is a decreasing function, and approaches negative infinity as \(x\) approaches zero from the right.

Key Concepts

Exponential Functions

Exponential functions display a fascinating characteristic where the variable is placed in the exponent. In the function form, this is typically represented as \( x = b^y \), where \( b \) is the base, and \( y \) is the exponent. Such functions grow or decay at rates proportional to their current value, a property that distinguishes them from linear functions.
An exponential graph has a distinct shape:

• It passes through the point (0,1), since \( b^0 = 1 \).
• For \( b > 1 \), the graph rises steeply to the right and flattens as \( y \) decreases.
• If \( 0 < b < 1 \), as in our case with \( b = \frac{1}{2} \), the function decreases, presenting a decay function.

The beauty of exponential functions shines in their rapid escalation or decline, making them applicable in fields ranging from compound interest calculations to population growth modeling.

Inverse Functions

Understanding inverse functions is crucial when delving into logarithmic and exponential relationships. An inverse function essentially reverses the operation of the original function, swapping the roles of input and output. If you have a function \( f \, \) and its inverse \( f^{-1} \,\), applying the inverse function \( f^{-1}(f(x)) \) will return you back to your original input \( x \).
In our specific scenario, the original exponential function \( x = (\frac{1}{2})^y \) transforms into its logarithmic inverse \( y = \log_{\frac{1}{2}} x \). Here:

• The \( y \) values of the exponential function become the \( x \) values of the logarithmic function.
• Similarly, the \( x \) values become the \( y \) values.

Recognizing these swaps aids significantly when graphing, as it directly informs which points lie on the logarithmic graph by utilizing the exponential graph's points.

Graphing Techniques

Graphing techniques for functions, especially logarithmic ones, call for methodical approaches. To ensure accuracy and understanding, one must grasp a few core principles:
Start with identifying the critical points from the exponential graph. This step aids in plotting because exponential functions serve as a mirror into which the logarithmic ones reflect.
Plot these transformed points, such as \((-2, 4)\) and \((0, 1)\), by swapping the \( x \) and \( y \) coordinates. Consequently, each point on the exponential function corresponds directly to a point on the logarithmic graph.

• Remember, the logarithmic graph will appear as if it's the reflection of the exponential graph across the line \( y = x \).
• The logarithmic graph descends, showing it is a decreasing function since our base is between 0 and 1, specifically half.

These techniques not only simplify the task of graphing but also boost comprehension of the intrinsic links between exponential and logarithmic functions.