Problem 64
Question
Use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-log plot. $$ y=6 x^{-1} $$
Step-by-Step Solution
Verified Answer
The linear relationship is \( \ln y = -\ln x + \ln 6 \) on a log-log plot with a slope of \(-1\).
1Step 1: Express in Logarithmic Form
Start with the given equation \( y = 6 x^{-1} \). Rewrite the equation in terms of natural logarithms to facilitate linearization: \( \ln y = \ln (6 x^{-1}) \). Use the logarithm property \( \ln(a \cdot b) = \ln a + \ln b \) to break it down: \( \ln y = \ln 6 + \ln x^{-1} \). Further simplify using \( \ln x^{-1} = -\ln x \) to get \( \ln y = \ln 6 - \ln x \).
2Step 2: Identify the Linear Equation
The expression \( \ln y = \ln 6 - \ln x \) can be rewritten in the form of a linear equation \( \ln y = -\ln x + c \), where the slope \( m = -1 \) and the y-intercept \( c = \ln 6 \). This is a linear relationship between \( \ln y \) and \( \ln x \).
3Step 3: Plot the Linear Relationship on a Log-Log Plot
On a log-log plot, plot \( \ln y \) (vertical axis) against \( \ln x \) (horizontal axis). The resulting graph will be a straight line with a slope of \(-1\) and a y-intercept of \( \ln 6 \). This visually confirms the linear relationship between the logarithmic variables.
Key Concepts
LinearizationLog-Log PlotGraphing Techniques
Linearization
Linearization is an essential mathematical technique used to simplify complex relationships into a more manageable, linear form. This process involves transforming a non-linear equation into a linear one, making it easier to analyze and understand.
When we talk about linearization in the context of the given exercise, we deal with logarithmic transformations. This involves applying logarithms to both sides of an equation. For instance, taking the equation, \( y = 6 x^{-1} \), we use the natural logarithm: \( \ln y = \ln (6 x^{-1}) \). This transformation helps us express the relationship in a linear form.
By using the properties of logarithms, namely that \( \ln(a \cdot b) = \ln a + \ln b \), we break down the components of the equation, translating products into sums. Therefore, \( \ln y = \ln 6 - \ln x \). This formula reflects a linear relationship where \( \ln y \) is expressed in terms of \( \ln x \). The slope of this linear equation is \(-1\), and its y-intercept is \( \ln 6 \). Linearization, hence, helps detect straightforward relationships within more complex data sets.
Understanding this technique is crucial for analyzing data trends, simplifying computations, and making predictions.
When we talk about linearization in the context of the given exercise, we deal with logarithmic transformations. This involves applying logarithms to both sides of an equation. For instance, taking the equation, \( y = 6 x^{-1} \), we use the natural logarithm: \( \ln y = \ln (6 x^{-1}) \). This transformation helps us express the relationship in a linear form.
By using the properties of logarithms, namely that \( \ln(a \cdot b) = \ln a + \ln b \), we break down the components of the equation, translating products into sums. Therefore, \( \ln y = \ln 6 - \ln x \). This formula reflects a linear relationship where \( \ln y \) is expressed in terms of \( \ln x \). The slope of this linear equation is \(-1\), and its y-intercept is \( \ln 6 \). Linearization, hence, helps detect straightforward relationships within more complex data sets.
Understanding this technique is crucial for analyzing data trends, simplifying computations, and making predictions.
Log-Log Plot
A log-log plot is a graphing method used to visualize data that spans several orders of magnitude and exhibits power-law relationships. In a log-log plot, both the x-axis and the y-axis are on logarithmic scales. This method is particularly useful when dealing with exponential decay or growth, as seen in our problem.
For instance, when we plotted \( \ln y \) against \( \ln x \) from the equation \( \ln y = -\ln x + \ln 6 \), the log-log plot turns a non-linear curve into a straight line. On the graph, we observe a linear trend which represents the power-law relationship between the original variables, \( y \) and \( x \).
The straight line in the log-log plot signifies that a consistent ratio exists between these variables. The slope of this line tells us about the nature of this relationship. In our problem, the slope is \(-1\), indicating an inverse variation. This means as \( x \) increases, \( y \) decreases and vice versa, maintaining a specific power relationship.
Overall, a log-log plot is a powerful tool simplifying visualization, making it easier to interpret complex relationships by linearizing them.
For instance, when we plotted \( \ln y \) against \( \ln x \) from the equation \( \ln y = -\ln x + \ln 6 \), the log-log plot turns a non-linear curve into a straight line. On the graph, we observe a linear trend which represents the power-law relationship between the original variables, \( y \) and \( x \).
The straight line in the log-log plot signifies that a consistent ratio exists between these variables. The slope of this line tells us about the nature of this relationship. In our problem, the slope is \(-1\), indicating an inverse variation. This means as \( x \) increases, \( y \) decreases and vice versa, maintaining a specific power relationship.
Overall, a log-log plot is a powerful tool simplifying visualization, making it easier to interpret complex relationships by linearizing them.
Graphing Techniques
Graphing techniques are vital for data analysis and interpretation, as they allow us to visualize patterns and relationships in data sets. Logarithmic transformations coupled with proper graphing techniques can simplify these visualizations.
When transforming an equation and plotting it on a log-log scale, one effectively unravels hidden linear relationships. For our exercise \( y = 6 x^{-1} \), the log transformation revealed the linear relationship \( \ln y = -\ln x + \ln 6 \) which can be plotted as a straight line. This is a fundamental aspect of graphing techniques used in analyzing mathematical equations.
Some key takeaways when applying graphing techniques:
When transforming an equation and plotting it on a log-log scale, one effectively unravels hidden linear relationships. For our exercise \( y = 6 x^{-1} \), the log transformation revealed the linear relationship \( \ln y = -\ln x + \ln 6 \) which can be plotted as a straight line. This is a fundamental aspect of graphing techniques used in analyzing mathematical equations.
Some key takeaways when applying graphing techniques:
- Identify which transformations, like logarithmic transformations, can linearize the relationship.
- Use log-log plots to visually confirm these relationships through straightforward graphical representation.
- Recognize the implications of the graph’s slope and intercept to interpret the data thoroughly.
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