Problem 43 - 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 rela\mathrm{\\{} t i o n s h i p . ~ ( T h e ~ o r i g i n a l ~ \(x-y\) coordinates are given.) $$ \left(x_{1}, y_{1}\right)=(0,5),\left(x_{2}, y_{2}\right)=(3,1) $$

Step-by-Step Solution

Answer

The functional relationship is \( y = 5^{1-x/3} \).

Step 1: Understand the Plot Type

The problem states that when \( \log y \) is plotted against \( x \), the result is a straight line. This is known as a log-linear plot, where the y-axis is in a logarithmic scale, and the x-axis is in a linear scale.

Step 2: Calculate the Logarithm of y-values

Calculate \( \log y \) for each given point. For \( y_1 = 5 \), \( \log y_1 = \log 5 \). For \( y_2 = 1 \), \( \log y_2 = \log 1 = 0 \).

Step 3: Write the Points for the Log-Linear Plot

Substitute the logarithmic values of y into the coordinates. This results in the points:1. \( (x_1, \log y_1) = (0, \log 5) \)2. \( (x_2, \log y_2) = (3, 0) \)

Step 4: Determine the Equation of the Line

The equation of a line is \( y = mx + c \). We need to find \( m \) (the slope) and \( c \) (the y-intercept).The slope \( m \) is calculated as follows:\[m = \frac{\log y_2 - \log y_1}{x_2 - x_1} = \frac{0 - \log 5}{3 - 0} = \frac{-\log 5}{3}\]Since the line passes through \((0, \log 5)\), the y-intercept \( c = \log 5 \).

Step 5: Write the Final Linear Equation

Substitute the slope and y-intercept into the line equation:\[\log y = -\frac{\log 5}{3}x + \log 5\]

Step 6: Express as a Functional Relationship

Rearrange the equation to express \( y \) as a function of \( x \):\[\log y = (-\frac{1}{3} \log 5) x + \log 5 \y = 10^{((-\frac{1}{3} \log 5) x + \log 5)} \\]Using properties of logarithms, this simplifies to:\[y = 5 \times (5^{(-1/3)})^x\] Therefore, \( y = 5^{1-x/3} \).

Key Concepts

Logarithmic Scale

A logarithmic scale involves plotting data in a way where each mark on the scale is a multiple of the previous one. Instead of each step on the scale representing an equal increase in linear terms, it represents a power increase. This means that each unit increase on a logarithmic scale represents multiplying the previous value by a constant factor. This is particularly useful in graphs where the data covers a wide range of values, as it can make it easier to visualize data that grows exponentially.
For instance:

• If using a base-10 logarithmic scale, moving from 1 to 2 on this scale represents a tenfold increase in actual value.
• This property helps to turn exponential growth into linear trends, which are easier to interpret.

In a log-linear plot, the y-axis is logarithmic, while the x-axis remains linear. This approach is used when the relationship between variables is multiplicative or exponential.

Linear Equation

Linear equations are equations that form a straight line when graphed. They frequently appear in different forms, such as the slope-intercept form, which is expressed as \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.
To understand the linear relationship in the given problem, we calculated the slope \( m \) and y-intercept \( c \) using our given points on the log-linear plot:

• The slope \( m \) is derived from the change in \( \log y \) over the change in \( x \), showing the steepness and direction of the line.
• The y-intercept \( c \) is where the line crosses the y-axis, in this context, it represents \( \log 5 \).

The equation \( \log y = -\frac{\log 5}{3}x + \log 5 \) represents our line on the log-linear plot. It's essential to understand that the straight line on this specific graph indicates an exponential relationship when considering the logarithmic transformation of \( y \).

Functional Relationship

In mathematics, a functional relationship expresses one quantity as a function of another. In this scenario, we explore how \( y \) depends on \( x \) in the transformed space. After solving the linear equation on the log-linear plot, we can reinterpret it back in terms of the original variables.
By expressing \( y = 10^{((-\frac{1}{3} \log 5) x + \log 5)} \), the equation indicates how the value of \( y \) changes with \( x \) when considering the log transformation. More clearly:

• This relationship implies that \( y \) changes with \( x \) in an exponential manner since it involves powers of 5.
• The expression simplifies to \( y = 5^{1-x/3} \), demonstrating the multiplicative effect \( x \) has as an exponent on the base 5.

Understanding these changes underscores the core idea that in exponential relationships, small changes in \( x \) result in proportionally larger adjustments in \( y \), which become linear when logged.