Problem 33 - College Algebra

Question

For the following exercises, refer to Table 8. $$\begin{array}{ccccccc}{x} & {1} & {2} & {3} & {4} & {5} & {6} \\ {f(x)} & {555} & {383} & {307} & {210} & {158} & {122}\end{array}$$ Write the exponential function as an exponential equation with base e.

Step-by-Step Solution

Answer

The exponential equation is \( f(x) = 555e^{-0.371x} \).

Step 1: Understanding the Form of an Exponential Function

The general form of an exponential function with base \( e \) is \( f(x) = ae^{bx} \), where \( a \) is the initial amount and \( b \) is the rate of change.

Step 2: Identifying the Initial Value \( a \)

From the table, we identify the initial value \( f(1) = 555 \). Therefore, when \( x = 1 \), the expression for the exponential function becomes \( f(1) = ae^{b} = 555 \).

Step 3: Calculating the Rate of Change \( b \)

To simplify the calculation of \( b \), take two points from the table. Let's use \( f(1) = 555 \) and \( f(2) = 383 \). Set up the equation for both:\[ 555 = ae^{b} \]\[ 383 = ae^{2b} \].

Step 4: Solving for \( b \)

Divide the second equation by the first to eliminate \( a \):\[ \frac{383}{555} = e^{b} \]Calculate \( \frac{383}{555} \approx 0.690 \). Taking the natural logarithm on both sides, we find:\[ b \approx \ln(0.690) \].

Step 5: Calculating \( b \)

Compute \( b \) from the previous step:\[ b \approx \ln(0.690) \approx -0.371 \].

Step 6: Constructing the Equation

Substitute back \( a = 555 \) (from \( f(1) = ae^{b} \) where we know \( e^{b} = \frac{383}{555} \)) and \( b = -0.371 \) into the exponential function:\[ f(x) = 555e^{-0.371x} \].

Key Concepts

Exponential Function

In mathematics, an exponential function is one where the variable appears in the exponent. It generally has the form \( f(x) = a \times b^x \). When the base is the mathematical constant \( e \) (approximately equal to 2.718), the function becomes \( f(x) = ae^{bx} \), where:

• \( a \) is the initial value, which dictates where the function starts when \( x = 0 \).
• \( e \) is the base of the natural logarithm.
• \( b \) represents the rate of change.

This form of function is highly valuable because it models growth and decay processes, like population growth or radioactive decay. Understanding exponential functions is crucial for analyzing processes that change rapidly or have compound growth over time.

Natural Logarithm

The natural logarithm, represented as \( \ln(x) \), is the logarithm to the base \( e \). Unlike common logarithms which use base 10, the natural logarithm uses the base \( e \), which is a special number in mathematics due to its unique properties.The natural logarithm is useful in finance, biology, and many other fields because it helps in understanding growth and change in natural processes. For the problem at hand, the natural logarithm helps us to solve for the rate of change \( b \) by taking the logarithm of both sides of the equation derived from matching two points on the exponential curve, and simplifying.Using the natural logarithm allows us to linearize exponential relationships, making complex growth processes easier to handle analytically.

Rate of Change

The rate of change in an exponential function, represented by \( b \) in the equation \( f(x) = ae^{bx} \), defines how quickly or slowly a quantity increases or decreases as \( x \) changes. This rate can tell us whether the function is growing or decaying:

• A positive \( b \) indicates growth, meaning the function values increase as \( x \) increases.
• A negative \( b \) represents decay, where the function values decrease as \( x \) increases.

To calculate \( b \), you can use data points from the function as seen in the step-by-step solution. In this example, the calculations involved comparing the values \( f(1) = 555 \) and \( f(2) = 383 \) using logarithms. This process allowed us to quantify the rapidity of change in the function's context.

Initial Value

The initial value in an exponential function is denoted as \( a \), representing the starting value when \( x = 0 \). It sets a benchmark from which growth or decay begins.In the context of the table and exercise above, the initial value is identified as \( f(1) = 555 \), which means the function value at \( x = 1 \) was used as \( a \). This step simplifies calculations because it provides a known anchor point, making it easier to solve for other unknowns in the function.Understanding the initial value helps contextualize data and ensures that any growth or decay is always related to a meaningful starting point. In practical terms, knowing the initial value allows you to predict future values or backtrack to past values effectively.