Problem 66 - Precalculus : Building Concepts and Connections

Question

In Exercises \(65-68,\) use a graphing utility with a decimal window. Graph \(f(x)=\log 0.1 x\) and \(g(x)=\log x\) on the same set of axes. Explain the relationship between the two graphs in terms of the properties of logarithms.

Step-by-Step Solution

Answer

The graph of \(f(x)=\log 0.1 x\) is a horizontal stretch of the graph of \(g(x)=\log x\). This is because the multiplier 0.1, smaller than 1, in the logarithm function results in the stretching of the graph.

Step 1: Plot the first function

Using a graphing utility, plot the first function \(f(x)=\log 0.1 x\). With a decimal window, the graph will show how the function behaves for different values of x.

Step 2: Plot the second function

Next, plot the second function \(g(x)=\log x\) onto the same set of axes. This will give a clear image of how the two functions interact.

Step 3: Compare the two functions

Examine the two graphs. Pay attention to the shape, location, and directional changes in the graphs. This will help understand the impact of the multiplier 0.1 in the logarithm function.

Step 4: Understand the properties of logarithms

The transformation from \(g(x)=\log x\) to \(f(x)=\log 0.1 x\) is a horizontal stretch. This is due to the properties of logarithms that state that a multiplication inside the logarithm function will translate into a stretch or a compression on the graph. In this case, \(0.1x\) is causing a stretch because 0.1 is less than 1.

Key Concepts

Graphing Utilities

Graphing utilities are tools that help us visualize mathematical functions by plotting their graphs on a coordinate plane. These utilities bring the functions to life, allowing us to see their behavior and how they interact with one another. A popular graphing utility is the calculator or computer software equipped with graphing capabilities like Desmos or GeoGebra.

When plotting functions such as logarithmic functions, these tools are extremely beneficial. They let us see the gradual rise of the curve as the input values change. In the case of the given exercise, using a graphing utility with a decimal window allows us to accurately plot both \(f(x) = \log 0.1 x\) and \(g(x) = \log x\) over a range of x-values. The utility will show you how each function behaves and enable you to compare them side by side on the same axes. This comparison helps visually observe the effects of changes made to the basic logarithmic function.

Properties of Logarithms

The properties of logarithms are key to understanding how logarithmic functions behave and transform. Here are a few fundamental properties:

• Product Property: \(\log(a \cdot b) = \log a + \log b\)
• Quotient Property: \(\log(\frac{a}{b}) = \log a - \log b\)
• Power Property: \(\log(a^b) = b \cdot \log a\)

These properties are essential when analyzing transformations.

In the exercise, the function \(f(x) = \log 0.1 x\) can be interpreted using these properties. Specifically, the multiplication inside the logarithm (\(0.1x\)) indicates a transformation. By properties of logarithms, \(\log 0.1 x\) can be expanded to \(\log 0.1 + \log x\). The constant \(\log 0.1\) effectively shifts the graph vertically. Understanding this relationship is crucial to differentiate between the effects of multiplying or adding values inside the logarithm.

Function Transformation

Function transformations adjust the position and shape of a graph in various ways. Two main types are horizontal and vertical transformations, and both can impact the way logarithmic graphs appear.

In this exercise, we focus on horizontal transformations. The function \(f(x) = \log 0.1 x\) is a horizontally stretched version of \(g(x) = \log x\). This happens because multiplying the inside of the logarithm by a factor less than 1 (in this case, 0.1) extends the graph horizontally. Essentially, each point on the graph of \(g(x)\) is translated outward on the x-axis, making the slope appear gentler compared to \(g(x)\).

Graphically, both functions rise from negative infinity towards positive infinity, but \(f(x)\) progresses more slowly away from the y-axis. Visualizing this transformation with a graphing utility makes understanding these changes more intuitive and demonstrates the geometric impact of internal multipliers within a logarithmic function.