Problem 53 - Algebra and Trigonometry with Analytic Geometry

Question

Change \(f(x)=1000(1.05)^{x}\) to an exponential function with base \(e\) and approximate the growth rate of \(f\).

Step-by-Step Solution

Answer

The exponential function with base \( e \) is \( f(x) = 1000 e^{0.04879x} \) and the growth rate is approximately 0.04879.

Step 1: Express Given Function in the Form of Base e

The given function is \( f(x) = 1000(1.05)^x \). To convert it to a form with base \( e \), recall the property of exponents: \( a^x = e^{x \ln a} \). Thus, rewrite \( (1.05)^x = e^{x \ln(1.05)} \).

Step 2: Substitute and Simplify

Substitute the expression from Step 1 into the given function: \( f(x) = 1000 \cdot e^{x \ln(1.05)} \). This rewriting keeps the function in terms of base \( e \).

Step 3: Calculate the Growth Rate

The growth rate for the exponential function \( e^{kx} \) is given by \( k \). Here, \( k = \ln(1.05) \). Calculate this using the natural logarithm: \( \ln(1.05) \approx 0.04879 \). This reflects the approximate growth rate of the function.

Key Concepts

Conversion to Base e

Exponential functions often involve different bases. Transforming the function's base is crucial for easier analysis and simplification. To convert any exponential function to base \(e\), use the property of exponents: \( a^x = e^{x \ln a} \). This property helps express any exponential base as an equation involving base \(e\). For instance, take the function \( f(x) = 1000(1.05)^x \). Here, the base is \(1.05\). The goal is to express this function in terms of \(e\). Using the aforementioned property, rewrite\( (1.05)^x = e^{x \ln(1.05)} \). This transforms the given function to \( f(x) = 1000 \cdot e^{x \ln(1.05)} \). This expression keeps the integrity of the function by using base \(e\), which is a natural choice for continuous growth processes.

Growth Rate Calculation

The growth rate is a key feature of exponential functions, telling us how quickly the function's values increase. For functions expressed as \( e^{kx} \), the growth rate is \(k\). This constant value \(k\) gives insights into the rate of change. To find \(k\) for the converted function \( f(x) = 1000 \cdot e^{x \ln(1.05)} \), recognize that \(k\) is \(\ln(1.05)\). Let's calculate this:

• Compute the natural logarithm of \(1.05\)
• \( \ln(1.05) \approx 0.04879 \)

This means the approximate growth rate of the function is \(0.04879\), reflecting how much the function grows for each unit increase in \(x\). Understanding this growth rate is essential for analyzing how fast or slow the process represented by the function occurs over time.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a mathematical concept frequently used in calculus and algebra. It specifically uses the base \(e\). This makes it important for solving problems related to growth and decay. The natural logarithm converts exponential functions into simpler linear forms, enabling easier manipulation and calculation. For example, when converting \((1.05)^x\) to a base \(e\), the natural logarithm \(\ln(1.05)\) was crucial. It's the bridge between different bases, facilitating conversions.

• \( \ln \) is used to calculate growth rates directly from base conversion.
• \( \ln(e) = 1 \), reinforcing its suitability for functions involving base \(e\).

Using \(\ln\) assists in tackling various problems involving exponential functions, both in growth and decay contexts. It provides a straightforward way to understand and calculate the behavior of exponential changes efficiently.