Problem 82 - Calculus for Biology and Medicine

Question

Use the indicated base to logarithmically transform each exponential relationship so that a linear relationship results. Then use the indicated base to graph each relationship in a coordinate system whose axes are accordingly transformed so that a straight line results. $$ y=3^{-x} ; \text { base } 3 $$

Step-by-Step Solution

Answer

Transform \( y = 3^{-x} \) with base 3 to \( \log_3(y) = -x \), a linear relationship.

Step 1: Understand the Exponential Relationship

The given exponential function is \( y = 3^{-x} \). Our goal is to transform this into a linear function using logarithms with base 3.

Step 2: Apply Logarithm with Indicated Base

To transform the relationship, apply a logarithm with base 3 to both sides: \( \log_3(y) = \log_3(3^{-x}) \).

Step 3: Simplify the Logarithmic Expression

Using the properties of logarithms, simplify \( \log_3(3^{-x}) \) as follows: \( \log_3(3^{-x}) = -x \cdot \log_3(3) \). Since \( \log_3(3) = 1 \), we have \( -x \). Thus, the equation simplifies to \( \log_3(y) = -x \).

Step 4: Resulting Linear Relationship

Now, the equation \( \log_3(y) = -x \) represents a linear relationship between \( \log_3(y) \) and \( x \). This is a straight line with slope \(-1\) and no intercept.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In our given problem, the exponential function is represented by:

• This function is written as: \( y = 3^{-x} \),
• Here, the base \( 3 \) is raised to the power of a negative value of \( x \).

This means that as \( x \) increases, the value of \( y \) decreases exponentially. Exponential functions can model real-world phenomena such as population growth, radioactive decay, and interest rates. Understanding these functions can help decipher how quantities change over time.

Linear Functions

Linear functions describe relationships where the dependent variable changes at a constant rate with respect to the independent variable. These functions are typically expressed in the form:

• \( y = mx + b \),
• Where \( m \) is the slope, and \( b \) is the y-intercept.

In the context of logarithmic transformation, like in the step-by-step solution, we turned the exponential function \( y = 3^{-x} \) into a linear function by applying a logarithm base 3, resulting in \( \log_3(y) = -x \).The slope here is \(-1\), which means with each increment of \( x \), the \( \log_3(y) \) decreases by \( 1 \). This transformation makes it easier to plot and interpret the relationship on a graph by translating it into a form that students are often more familiar with.

Properties of Logarithms

Logarithms are the inverse operations of exponentials, and understanding their properties is crucial to transforming exponential functions into linear ones. One key property used in the original solution is:

• \( \log_b(b^x) = x \), which implies that taking the logarithm of a number with its base results in the exponent.

In our example:

• \( \log_3(3^{-x}) = -x \cdot \log_3(3) \)
• Simplifies to \( -x \) because \( \log_3(3) = 1 \).

This simplification illustrates a fundamental logarithmic property that helps convert the non-linear form into a linear equation. By mastering these properties, students can efficiently perform transformations and understand the underlying structure of equations.