Problem 51 - Intermediate Algebra

Question

For Problems \(47-53\), graph each of the functions. Remember that the graph of \(f(x)=\log _{2} x\) is given in Figure 11.7. $$ f(x)=\log _{2} 2 x $$

Step-by-Step Solution

Answer

Graph \(f(x) = \log_{2}(x)\) and shift it up by 1 unit.

Step 1: Understand the function

We are given the function \(f(x) = \log_{2}(2x)\). This is a logarithmic function where the base is 2. Our goal is to graph this function based on the understanding of its components.

Step 2: Apply Logarithmic Properties

Recall the logarithmic property \( \log_b(mn) = \log_b m + \log_b n \). Applying this to our function, we get: \[ f(x) = \log_{2}(2x) = \log_{2}(2) + \log_{2}(x) \] Since \(\log_{2}(2) = 1\), it simplifies to: \[ f(x) = 1 + \log_{2}(x) \].

Step 3: Analyze the transformation

The function \(f(x) = \log_{2}(x)\) is shifted vertically by 1 unit upwards to get \(f(x) = 1 + \log_{2}(x)\). This means every point on the graph of \(\log_{2}(x)\) is moved up by 1 unit.

Step 4: Sketch the graph

Use the graph of \(f(x) = \log_{2}(x)\), which is a curve that passes through (1,0) and approaches the y-axis as a vertical asymptote. Shift this graph 1 unit upwards. This new graph will pass through (1,1). It maintains the same asymptotic behavior and shape.

Key Concepts

Graphing Logarithmic Functions

Graphing logarithmic functions can often seem tricky at first, but with some practice, it becomes intuitive. Let's break it down. First, remember that the basic logarithmic function, such as \( f(x) = \log_{2}(x) \), features several key characteristics:

• A vertical asymptote along the y-axis; the graph never touches or crosses this line.
• The curve passes through the point (1,0) since the log base of its own number equals 1.
• The graph increases slowly to the right of the y-axis, reflecting the logarithmic nature of the function.

When we graph more complex logarithmic functions, like \( f(x) = \log_{2}(2x) \), these characteristics form the foundation. Analyzing how transformations, like shifts, affect the graph helps us sketch accurate representations of these functions.

Logarithmic Properties

Understanding the key logarithmic properties is crucial to simplifying and modifying logarithmic functions. Consider the property \( \log_b(mn) = \log_b m + \log_b n \). This tells us that the logarithm of a product can be expressed as a sum of logarithms.
In our function \( f(x) = \log_{2}(2x) \), applying this property helps us break it down into simpler components: \[ f(x) = \log_{2}(2x) = \log_{2}(2) + \log_{2}(x) \] Here, \( \log_{2}(2) = 1 \) because the base is the same as the argument, simplifying our equation to: \[ f(x) = 1 + \log_{2}(x) \].
Using these properties not only aids in simplifying expressions but also in understanding deeper transformations that occur when graphing.

Function Transformation

Function transformation is a powerful tool to visualize changes in the basic graph and understand how modifications alter the function's behavior. In the case of our function, after applying logarithmic properties, we simplify it to \( f(x) = 1 + \log_{2}(x) \).

• This +1 indicates a vertical shift upwards by one unit.
• Every point of the original graph \( \log_{2}(x) \) is adjusted, while the vertical asymptote remains unchanged.

By mapping this transformation onto the graph, the point (1,0) of \( \log_{2}(x) \) becomes (1,1), and the shape of the graph is preserved.
Understanding transformation makes it easier to sketch modified functions on the same plane, offering a clear visual representation of changes.