Problem 9 - Algebra and Trigonometry

Question

Find the solution of the exponential equation, correct to four decimal places. $$ e^{1-4 x}=2 $$

Step-by-Step Solution

Answer

The solution is approximately 0.0767.

Step 1: Transform to Logarithmic Form

To solve the equation \(e^{1-4x} = 2\), we start by taking the natural logarithm of both sides. This step involves using the identity that if \(a = e^b\), then \(b = \ln(a)\). Hence, we have:\[1 - 4x = \ln(2)\]

Step 2: Rearrange the Equation for x

Now that we have \(1 - 4x = \ln(2)\), the next step is to solve for \(x\). Begin by isolating the \(-4x\) term:\[-4x = \ln(2) - 1\]

Step 3: Solve for x

To find \(x\), divide both sides of the equation by \(-4\):\[x = \frac{\ln(2) - 1}{-4}\]Calculate this expression to four decimal places.

Step 4: Calculate the Expression

Using a calculator, compute the value:\[x = \frac{\ln(2) - 1}{-4} \approx \frac{0.6931 - 1}{-4} \approx \frac{-0.3069}{-4} \approx 0.0767\]

Key Concepts

Natural Logarithms

Natural logarithms are logarithms with the base of Euler's number, denoted as \(e\), which is approximately 2.71828. They are represented as \(\ln(x)\), where \(\ln\) stands for the natural logarithm. These are particularly useful in problems involving growth, decay, and complex mathematical transformations because their base, \(e\), has special properties in calculus and mathematical analysis.
A primary property of the natural logarithm is that \(\ln(e) = 1\). This arises because any number raised to the power of one gives back the same number. Therefore, if you have an exponential expression like \(e^y\), taking the natural logarithm \(\ln(e^y)\) simply returns \(y\).

• This property forms the basis of solving exponential equations with natural logs, as it helps in converting an exponential equation into a linear one, simplifying the process.
• For example, converting \(e^b = a\) into \(b = \ln(a)\).

Understanding and manipulating natural logarithms is a cornerstone in calculus and helps unravel complex equations by reducing their base exponential form.

Solving Equations

When it comes to solving equations, especially exponential ones, the process usually involves isolating the variable of interest. For the equation \(e^{1-4x} = 2\), the goal is to solve for \(x\).
One popular method for tackling such equations is first using logarithms to simplify the equation. By employing logarithmic transformation, we can convert it into a linear equation.
Here's the typical step-by-step approach to solve these kinds of equations:

• Apply the logarithm (in this case, the natural logarithm) to both sides of the equation.
• Simplify the equation using the properties of logarithms.
• Rearrange the equation to isolate the variable \(x\).
• Calculate the numerical results as required by the problem prompt.

This systematic approach helps in breaking down potentially complicated exponential equations into simpler linear equations. By following these steps, you can tackle a variety of equation types, making solving equations an essential tool in your math toolkit.

Logarithmic Transformation

Logarithmic transformation is a powerful technique used in mathematics to simplify complex expressions, especially exponential equations. The transformation involves applying a logarithm, usually natural logarithm \(\ln\), to every term within an equation.
This process leverages the logarithmic identities, such as:

• If \(a = b^c\), then \(c = \log_b(a)\).
• For the natural log, if \(a = e^y\), then \(y = \ln(a)\).

Through these transformations, equations that initially seem daunting because of exponential terms, become a structured and manageable format.
In our example, \(e^{1-4x} = 2\), by taking the natural log of both sides, we convert the exponentiated term into a linear equation \(1 - 4x = \ln(2)\). This transformation dramatically reduces the complexity of solving the equation.
By transforming exponential equations into a linear format, it becomes straightforward to apply algebraic methods to find the variable of interest. Mastering logarithmic transformations allows you to tackle a variety of problems across different areas of mathematics and applied disciplines.