Problem 384 - Precalculus

Question

For the following exercises, use the logistic growth model \(f(x)=\frac{150}{1+8 e^{-2 x}}\) Rewrite \(f(x)=1.68(0.65)^{x}\) as an exponential equation with base \(e\) to five significant digits.

Step-by-Step Solution

Answer

The equation is \(f(x) = 1.68e^{-0.43078x}\).

Step 1: Understand the Form of Exponential Equation

The given equation is in the form of \(f(x) = ab^x\), which is an exponential function. We need to convert it into an exponential equation with base \(e\), which typically takes the form \(f(x) = ae^{kx}\).

Step 2: Express the Exponent in Base e

Since \(f(x) = 1.68(0.65)^x\), we can rewrite the base \(0.65\) in terms of \(e\). This means finding \(k\) such that \((0.65)^x = e^{kx}\).

Step 3: Solve for k Using the Natural Logarithm

To express \(0.65\) in terms of \(e\), take the natural logarithm: \(0.65 = e^{\ln(0.65)}\). Therefore, \((0.65)^x = e^{x\ln(0.65)}\), so \(k = \ln(0.65)\). Calculate \(k\):

Step 4: Calculate Natural Logarithm and Substitute k

Using a calculator, \(\ln(0.65) \approx -0.43078\). Hence, \(f(x) = 1.68e^{-0.43078x}\).

Step 5: Verify and Write the Final Exponential Equation

So, the equivalent exponential function with base \(e\) is \(f(x) = 1.68e^{-0.43078x}\) to five significant digits.

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where the variable appears in the exponent. These equations are generally in the form of \(f(x) = ab^x\), where \(a\) is a constant and \(b\) is the base of the exponent. Understanding exponential equations is crucial because they model real-world phenomena such as population growth, radioactive decay, and interest calculations.
In an exponential equation, the rate of growth or decay depends on the base \(b\). If \(b\) is greater than 1, the function represents exponential growth, while if \(b\) is less than 1, it represents exponential decay.
It is essential to be able to rewrite these equations in different forms, such as converting to base \(e\), which allows easier mathematical manipulation and analysis, especially in scientific contexts.

Natural Logarithm

The natural logarithm is a logarithm to the base \(e\). It is commonly denoted by \(\ln\). The number \(e\) is an irrational constant approximately equal to 2.71828. The natural logarithm is the inverse function of the exponential function with base \(e\).

• When you take the natural logarithm of a number, you determine the power to which \(e\) must be raised to equal that number.
• For instance, \(\ln(0.65)\) yields the exponent \(k\) when solving \(0.65 = e^k\).

The natural logarithm is vital when working with exponential growth models, as it helps Linearize the relationship between the variables involved.
It allows for solving exponential equations and understanding changes in growth rates. It's also widely used in calculus, particularly in integration and differentiation, due to its natural derivative and integral properties.

Base e

Base \(e\) is a crucial concept in higher mathematics, particularly in calculus and differential equations. The constant \(e\), approximately 2.71828, is the unique base of the natural logarithm, which makes calculations inherently easier when dealing with continuous growth or decay processes.

• Exponential functions with base \(e\) are of the form \(e^{kx}\), where \(k\) is a constant.
• Using \(e\) as a base simplifies the differentiation and integration of exponential functions.

When you express functions in terms of base \(e\), you utilize its unique mathematical properties for more streamlined computations.
Many natural processes, including the compounding of interest rates and population dynamics, are best modeled using exponents with base \(e\). By converting other bases to \(e\), such as the conversion in the step-by-step solution from \(b = 0.65\) to an equation with base \(e\), mathematical operations become more intuitive.