Problem 64
Question
Use a logarithmic transformation to find a linear relationship between the given quantities and graph the \mathrm{\\{} r e s u l t i n g ~ l i n e a r ~ r e l a t i o n s h i p ~ o n ~ a ~ l o g - l o g ~ p l o t . ~ $$ y=6 x^{-1} $$
Step-by-Step Solution
Verified Answer
Linear relationship on a log-log plot: \( \ln(y) = \ln(6) - \ln(x) \) with a slope of \(-1\).
1Step 1: Express the given equation in exponential form
The given equation is \( y = 6x^{-1} \). Rewrite it in a more exponential form: \( y = \frac{6}{x} \). This can also be written using negative exponents as \( y = 6x^{-1} \), which explicitly shows the power of \( x \).
2Step 2: Apply logarithms to both sides
Take the natural logarithm of both sides of the equation: \( \ln(y) = \ln(6x^{-1}) \). Utilize logarithmic properties to split this into simpler components: \( \ln(y) = \ln(6) + \ln(x^{-1}) \).
3Step 3: Simplify using logarithm properties
Using the power property of logarithms, simplify \( \ln(x^{-1}) \) as \( -\ln(x) \). Therefore, the equation becomes: \( \ln(y) = \ln(6) - \ln(x) \).
4Step 4: Recognize the linear form
The equation \( \ln(y) = \ln(6) - \ln(x) \) is of the form \( Y = b + mX \), where \( Y = \ln(y) \), \( X = \ln(x) \), \( b = \ln(6) \), and the slope \( m = -1 \). This implies a linear equation with intercept \( \ln(6) \) and slope \(-1\).
5Step 5: Graph the linear relationship on a log-log plot
On a log-log plot, plot \( \ln(y) \) against \( \ln(x) \). You should expect to see a straight line with a negative slope of \(-1\), intersecting the y-axis at \( \ln(6) \).
Key Concepts
Log-Log PlotLinear RelationshipExponential Form
Log-Log Plot
A log-log plot is a type of graph that makes it easier to identify power law relationships between variables. Instead of plotting regular axes, both the x and y axes use a logarithmic scale. When you graph points on a log-log plot, straight lines indicate a power law or polynomial relationship between the variables.
In this exercise, we are looking at the relationship expressed by the equation \(y = 6x^{-1}\). Through logarithmic transformation, the equation becomes \(\ln(y) = \ln(6) - \ln(x)\). When plotting \(\ln(y)\) against \(\ln(x)\) on a log-log plot, you should expect to see a straight line. This line, due to the negative sign, slopes downwards from left to right, confirming the inverse relationship between \(x\) and \(y\).
In this exercise, we are looking at the relationship expressed by the equation \(y = 6x^{-1}\). Through logarithmic transformation, the equation becomes \(\ln(y) = \ln(6) - \ln(x)\). When plotting \(\ln(y)\) against \(\ln(x)\) on a log-log plot, you should expect to see a straight line. This line, due to the negative sign, slopes downwards from left to right, confirming the inverse relationship between \(x\) and \(y\).
- Straight line: signifies power relationships
- Axes: Both are logarithmic
- Points on plot: Derived from taking \(\ln\) of original values
Linear Relationship
A linear relationship refers to a scenario where there is a consistent rate of change between two variables. In other words, if you graph these variables, you will get a straight line. We usually describe such a line using the linear equation formula \(Y = b + mX\), where \(m\) is the slope and \(b\) is the y-intercept.
In terms of logarithms, applying a logarithmic transformation to both sides of an equation might turn a nonlinear relationship into a linear one. For the equation \(\ln(y) = \ln(6) - \ln(x)\), it takes the form of \(Y = b + mX\) with \(Y = \ln(y)\), \(X = \ln(x)\), \(b = \ln(6)\), and \(m = -1\).
In terms of logarithms, applying a logarithmic transformation to both sides of an equation might turn a nonlinear relationship into a linear one. For the equation \(\ln(y) = \ln(6) - \ln(x)\), it takes the form of \(Y = b + mX\) with \(Y = \ln(y)\), \(X = \ln(x)\), \(b = \ln(6)\), and \(m = -1\).
- Constant slope: Reflects a constant rate of change
- Intercept \(b\): Value of \(Y\) when \(X\) is zero
- Negative slope: Shows inverse relationship
Exponential Form
The concept of exponential form is important in mathematical expressions, particularly those involving growth and decay. Here, we often describe our equations using exponents. In the equation given \(y = 6x^{-1}\), this means \(y = \frac{6}{x}\).
This formulation shows reciprocal action, where as \(x\) increases, \(y\) decreases proportionally. Converting such expressions to use the base of natural logarithms, say by using exponential notation, often goes hand in hand with using logarithmic transformations to simplify the relationship expressed through the negative or fractional exponent.
Recapping:
This formulation shows reciprocal action, where as \(x\) increases, \(y\) decreases proportionally. Converting such expressions to use the base of natural logarithms, say by using exponential notation, often goes hand in hand with using logarithmic transformations to simplify the relationship expressed through the negative or fractional exponent.
Recapping:
- Features negative exponent: Indicates inverse relationship
- Exponential notation: Facilitates recognizing and using power laws
- Fractional forms: Can simplify complex equations
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