Problem 54 - Calculus for Biology and Medicine

Question

use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-linear plot. $$ y=5^{-6 x} $$

Step-by-Step Solution

Answer

Transform the equation using the natural log: \( \ln(y) = -6 \ln(5) \cdot x \), which represents a straight line on a log-linear plot.

Step 1: Take the Natural Logarithm of Both Sides

To find a linear relationship, we need to transform the given equation. Start by taking the natural logarithm (log base e) of both sides of the given equation.\[ \ln(y) = \ln(5^{-6x}) \]

Step 2: Apply Logarithmic Rules

Apply the power rule of logarithms, which allows us to bring the exponent down in front of the logarithm. This rule states that \( \ln(a^b) = b \cdot \ln(a) \).\[ \ln(y) = -6x \cdot \ln(5) \]

Step 3: Simplify the Equation

The equation \( \ln(y) = -6x \cdot \ln(5) \) is now a linear equation in the form \( \ln(y) = mx + c \), where the slope \( m = -6 \ln(5) \) and the y-intercept \( c = 0 \).This simplifies to a linear equation:\[ \ln(y) = -6 \ln(5) \cdot x \]

Step 4: Prepare for Plotting on a Log-Linear Plot

On a log-linear plot of \( x \) vs. \( \ln(y) \), \( \ln(y) \) will appear on the vertical axis while \( x \) is on the horizontal axis.Given the equation \( \ln(y) = -6 \ln(5) \cdot x \), we have a straight line with a negative slope of \(-6 \ln(5)\) intersecting at the origin.

Key Concepts

Logarithmic Functions

Logarithmic functions help us solve complex equations more easily. They are the inverse of exponential functions. This means that while an exponential function grows rapidly, a logarithm brings it down to a more manageable level by finding which power the base has to be raised to, to get that number. For example, in the equation \( y = 5^{-6x} \), direct solutions can be tricky. But applying logarithms simplifies it greatly. When you take the natural logarithm (log base \( e \)) of both sides of the equation it transforms into a form that is much easier to interpret and graph.
This transformation is key in mathematical modeling, especially in fields like finance and natural sciences, where the translation of exponential growth into a linear form through logarithms reveals clearer insights.

Linear Relationship

A linear relationship between two variables is represented by a straight line on a graph. The simplicity of linear equations allows us to clearly understand how a change in one variable will affect another. In the transformed equation, \( \ln(y) = -6 \ln(5) \cdot x \), we see a linear form where \( \ln(y) \) is directly related to \( x \) via the slope. The slope, in this case, \( -6 \ln(5) \), shows the rate of change. Additionally, this setup resembles the linear equation formula \( y = mx + c \), with \( c \) equals zero here, indicating that the line passes through the origin.
This linearization allows us to visualize and interpret the relationship more easily, as plotting \( \ln(y) \) against \( x \) results in a straight line. Thus, recognizing and transforming exponential relationships into linear ones with logarithmic transformation greatly aids in data analysis.

Natural Logarithms

Natural logarithms use 'e' (approximately 2.718) as their base, which makes them particularly useful in various scientific calculations. In the realm of calculus and higher mathematics, they are preferred due to their natural properties and because they relate smoothly to growth and decay patterns in nature.

• The natural logarithm of a number tells us what power we must raise \( e \) to obtain that number. This property simplifies many calculus operations.
• When we take the natural logarithm of both sides of the equation \( y = 5^{-6x} \), it helps convert the exponential relationship into a form \( \ln(y) = -6 \ln(5) \cdot x \), which is linear.

Natural logarithms play an essential role in converting complex exponential growth or decay into straight lines. Thanks to this concept, mathematicians and scientists can better predict and understand natural phenomena. This explains why natural logarithms are a crucial component in any log-based transformation.