Problem 23 - Algebra and Trigonometry

Question

Find the solution of the exponential equation, correct to four decimal places. $$ \frac{50}{1+e^{-x}}=4 $$

Step-by-Step Solution

Answer

The solution to the equation is approximately \(-2.4423\).

Step 1: Express the Given Equation to Highlight the Exponential Component

The given equation is \( \frac{50}{1+e^{-x}} = 4 \). To isolate the exponential term, multiply both sides by \(1 + e^{-x}\): \(50 = 4(1 + e^{-x})\).

Step 2: Distribute and Isolate the Exponential Term

Expand the right side of the equation: \(50 = 4 + 4e^{-x}\). Now, subtract 4 from both sides to isolate the term with \(e^{-x}\): \(46 = 4e^{-x}\).

Step 3: Solve for the Exponential Term

Divide both sides by 4 to solve for \(e^{-x}\): \(e^{-x} = \frac{46}{4}\). Simplify the fraction: \(e^{-x} = 11.5\).

Step 4: Take the Natural Logarithm

To solve for \(x\), take the natural logarithm of both sides: \(-x = \ln(11.5)\).

Step 5: Solve for x

Solve for \(x\) by multiplying both sides by -1: \(x = -\ln(11.5)\).

Step 6: Calculate the Value of x

Use a calculator to compute \(-\ln(11.5)\). Ensure the answer is precise to four decimal places: \(x \approx -2.4423\).

Key Concepts

Natural Logarithm

In mathematics, the natural logarithm, denoted as \(ln\), is a powerful tool for solving equations involving exponential terms. The term 'natural' comes from the base of the logarithm, which is the number \(e\), approximately equal to 2.71828. Understanding the natural logarithm is essential for transforming complex exponential expressions into more manageable algebraic forms. It is particularly useful in solving equations where the unknown variable appears as the power of \(e\).

• The natural logarithm function helps to 'undo' the exponential function, allowing us to solve for the variable.
• It transforms multiplication into addition, which simplifies many mathematical operations.
• The relationship \(e^x = y\) can be rewritten as \(x = \ln(y)\), making it easier to isolate \(x\).

Here, the natural logarithm was used to convert the equation \(e^{-x} = 11.5\) into a more straightforward form, \(-x = \ln(11.5)\), which facilitated the solution of the problem.

Exponential Functions

Exponential functions are a vital part of mathematics, characterized by the constant \(e\) raised to the power of a variable. These functions can grow rapidly and are used to describe many real-world phenomena.In the expression \(e^x\), \(e\) is a mathematical constant, and \(x\) is the exponent. When dealing with exponential functions:

• The function \(e^x\) describes unlimited growth, while \(e^{-x}\) indicates decay.
• Exponential functions are useful for modeling population growth, financial investments, and more.
• The inverse of an exponential function is a logarithmic function, which helps in solving exponential equations.

In the given problem, \(e^{-x}\) plays a central role, showing exponential decay as part of the equation \(\frac{50}{1+e^{-x}}=4\). Solving this type of problem often involves algebraic manipulation and taking logarithms to simplify.

Equation Solutions

Solving equations, particularly exponentials and logarithms, requires methodological steps for accurate solutions. An exponential equation like \(\frac{50}{1+e^{-x}}=4\) necessitates isolating the exponential term first.Here's a recap of how to approach such equations effectively:

• Identify and isolate the exponential term through algebraic manipulation.
• Use distributive laws, as seen when expanding \(50 = 4(1+e^{-x})\) to \(50 = 4 + 4e^{-x}\).
• Rearrange terms to isolate the exponential, as in \(46 = 4e^{-x}\).
• Once isolated, take natural logarithms to break down the exponent, turning the equation into a linear form.
• Calculate using logarithmic values to find the unknown variable accurately.

Eventually, by computing \(-\ln(11.5)\), the solution for \(x\) was found to be approximately \(-2.4423\). The clarity of each step helps ensure precision in mathematical problem-solving.