Problem 94 - College Algebra

Question

Graph \(y=\log x, y=\log (10 x),\) and \(y=\log (0.1 x)\) in the same viewing rectangle. Describe the relationship among the three graphs. What logarithmic property accounts for this relationship?

Step-by-Step Solution

Answer

The property of logarithms that accounts for this relationship is the transformation property that shifts the graph left or right depending on whether the logarithm's argument is multiplied by a value greater or less than 1. Graphing shows \(y=\log x\) as the standard log function, \(y=\log (10x)\) shifted left, and \(y=\log (0.1 x)\) shifted right.

Step 1: Graph First Function

First, graph the function \(y=\log x\). This function represents the basic log function, exhibiting a vertical asymptote at \(x=0\) and crossing the x-axis at \(x=1\).

Step 2: Graph Second Function

Graph the function \(y=\log (10x)\). Notice the graph is translated to the left relative to the original log function. This is because multiplying the input of the logarithm by 10, the function graph shifts toward the origin.

Step 3: Graph Third Function

Lastly, graph the function \(y=\log (0.1x)\). The graph moves to the right relative to the original function. This is happening because we are multiplying the input by less than one, which causes a rightward shift.

Step 4: Explain the Relationship

The graphs demonstrate that multiplying the input of the logarithm by a number greater than 1 will translate the graph to the left, and multiplying by a number less than 1 will translate the graph to the right.

Key Concepts

Logarithmic Graph Transformations

Understanding how logarithmic graphs transform is essential when graphing different functions. A basic logarithmic graph, such as the graph of \(y = \log x\), has a characteristic shape: it increases slowly and crosses the x-axis at the point \((1, 0)\). This pattern helps in identifying shifts due to transformations.

Now, let's see what happens when we apply transformations by multiplying the input value (x) by a coefficient like 10 or 0.1. When we graph \(y = \log (10x)\), the graph shifts to the left. This leftward shift occurs because inputs are scaled, making each x-value reach what used to be the same output at a smaller x. Essentially, multiplying by 10 means needing a smaller x-value to get the same log result. Conversely, for \(y = \log (0.1x)\), the graph shifts to the right as it now takes a larger x-value to achieve the same logarithmic result. These shifts due to scaling are an important property of logarithmic transformations.

• A multiplier greater than 1 causes a leftward shift.
• A multiplier less than 1 results in a rightward shift.

Properties of Logarithms

Logarithms are useful and versatile in mathematics due to their unique properties which help simplify complex calculations. One crucial property is how they translate multiplication into addition: \(\log (ab) = \log a + \log b\). This property reveals how transforming the input of a logarithmic function affects its graph.

In the context of graphical transformations, the multiplicative change of the input, like 10 in \(y = \log (10x)\), leverages the property \(\log (10x) = \log 10 + \log x\). The \(\log 10\) acts as a constant shift to the left. Similarly, with \(\log (0.1x)\), equivalently \(\log (\frac{1}{10}x) = -\log 10 + \log x\), there is a rightward shift caused by \(-\log 10\).

• \(\log (ab) = \log a + \log b\) is foundational for understanding transformations.
• Constant terms from properties shift the graph.

Graphing Functions

Graphing functions systematically involves understanding their unique characteristics and how transformations affect their shape. Starting with the basic form, such as \(y = \log x\), consider this as your baseline or reference graph.

When graphing transformations such as \(y = \log (10x)\) and \(y = \log (0.1x)\), it can assist immensely to apply transformations step-by-step. Moving the graph of a basic log to the left or right accurately is crucial. Always look for key points like intercepts or asymptotes to gain intuitive understanding.

Practicing with many functions and transformations will build a strong grasp of how logs behave spatially. Begin with straightforward graphs, note major shifts, and ensure understanding of why shifts occur.

• Identify the basic form before applying transformations.
• Mistake prevention: Start with step-by-step graphical manipulation.
• Focus on key points: intercepts and asymptotes guide the shape.