Problem 44 - Calculus for Biology and Medicine

Question

When log \(y\) is graphed as a function of \(x, a\) straight line results. Graph straight lines, each given by two points, on a log-linear plot, and determine the functional relationship. (The original \(x-y\) coordinates are given.) $$ \left(x_{1}, y_{1}\right)=(-1,4),\left(x_{2}, y_{2}\right)=(2,8) $$

Step-by-Step Solution

Answer

The relationship is \(y = 10^{0.1003x + 0.7023}\).

Step 1: Convert the Coordinates for the Logarithmic Scale

To determine the functional relationship, we first take the logarithm base 10 of the given y-values. For point \((x_1, y_1) = (-1, 4)\), convert \(y_1\) to log scale: \(\log_{10}(y_1) = \log_{10}(4)\). For \((x_2, y_2) = (2, 8)\), convert \(y_2\) similarly: \(\log_{10}(y_2) = \log_{10}(8)\). You can use a calculator to find these values.

Step 2: Calculate Logarithmic Values

Calculate the logarithm of each y-value. For \(y_1 = 4\), \(\log_{10}(4) \approx 0.602\). For \(y_2 = 8\), \(\log_{10}(8) \approx 0.903\). So the converted points on the log-linear plot are \((-1, 0.602)\) and \((2, 0.903)\).

Step 3: Determine the Slope of the Line

The slope \(m\) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute the converted values: \(m = \frac{0.903 - 0.602}{2 - (-1)} = \frac{0.301}{3} = 0.1003\).

Step 4: Find the Equation of the Line

Using the slope-point form of a line, \(y - y_1 = m(x - x_1)\), select one of the points, say \((-1, 0.602)\), and use the slope \(m = 0.1003\): \( y - 0.602 = 0.1003 (x + 1) \). Simplifying, we get:\( y = 0.1003x + 0.7023 \).

Step 5: Derive the Functional Relationship

Since \(y\) represents \(\log_{10}(y)\) in our equation from Step 4, rewrite it back into exponential form to find the function of \(y\) with respect to \(x\). This converts to: \(\log_{10}(y) = 0.1003x + 0.7023\) which gives \(y = 10^{0.1003x + 0.7023}\). This is the original relationship between \(x\) and \(y\) we were looking for.

Key Concepts

Slope Calculation on a Log-Linear Plot

To determine the functional relationship between two points on a log-linear plot, calculating the slope is an essential step. The slope tells us how steep the line is and helps in defining the exact relationship. In a typical situation, we have two points, say \((x_1, y_1)\) and \((x_2, y_2)\). The slope \(m\) is calculated using the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]For a log-linear plot, we use the transformed logarithmic values of \(y\) in this formula. Once we replace the original \(y\)-values with their logarithmic equivalents, we can accurately find the slope of the plotted line. This slope is crucial because it directly influences the functional form of the relationship between \(x\) and the original \(y\). Think of it as the key to unlock not just a line's direction, but also the interplay of variables in exponential relationships.

Understanding Exponential Functions

Exponential functions are vital when dealing with log-linear plots, especially because they represent the relationship between variables where one is a constant raised to the power of another variable. These functions are written as:\[y = a \cdot b^x\]where \(a\) is a constant, \(b\) is the base of the exponent, and \(x\) is the variable exponent.In the context of log-linear graphs, once you have determined the relationship as log-linear, converting back to an exponential function is crucial. When you find an equation like \(\log_{10}(y) = 0.1003x + 0.7023\), you can turn it into an exponential function by taking the base 10 antilogarithm:\[y = 10^{0.1003x + 0.7023}\]This conversion shows how changes in \(x\) produce multiplicative changes in \(y\), emphasizing the rapid growth characteristic of exponential functions.

Log-Linear Graphing Techniques

Graphing on a log-linear scale involves specific techniques distinct from standard linear graphs. In a log-linear plot, the horizontal axis follows a linear scale, while the vertical axis uses a logarithmic scale. This setup is particularly useful for analyzing data that spans several orders of magnitude.

• Identify Logarithmic Values: Begin by transforming the \(y\)-values into their logarithmic forms. This sets the stage for accurate graph representation.


• Plot Transformed Points: Use the linear \(x\)-values and their corresponding logarithmic \(y\)-values to plot the points. The main advantage here is the visualization of exponential relationships as straight lines.


• Draw the Best Fit Line: Once the points are plotted, connect them to form a line. The straight line on a log-linear graph indicates a consistent exponential relationship between the variables.

Utilizing these methods allows one to appreciate data trends more clearly, providing a nuanced understanding of how variables relate in exponential contexts.