Problem 11 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The solution is approximately \( x \approx 0.0767 \).

Step 1: Write the Equation

Given the equation, we need to solve it for \(x\): \[ e^{1-4x} = 2 \]

Step 2: Apply the Natural Logarithm

To eliminate the exponential function, take the natural logarithm of both sides: \[ \ln(e^{1-4x}) = \ln(2) \] The property of logarithms allows us to simplify this to: \[ 1-4x = \ln(2) \]

Step 3: Isolate the Variable Term

Subtract 1 from both sides to isolate the term with \(x\): \[ -4x = \ln(2) - 1 \]

Step 4: Solve for x

Divide both sides by \(-4\) to solve for \(x\): \[ x = \frac{\ln(2) - 1}{-4} \]

Step 5: Calculate and Round the Solution

Using a calculator, we approximate \( x \): \[ x \approx \frac{0.6931 - 1}{-4} = \frac{-0.3069}{-4} \approx 0.0767 \]

Key Concepts

Natural Logarithms

Natural logarithms are a specific type of logarithm where the base is the mathematical constant \( e \), approximately equal to 2.71828. They are commonly denoted as \( \ln \). When solving equations that involve exponential functions with base \( e \), natural logarithms serve as a pivotal tool.

• They allow us to transform exponential equations into linear ones that are easier to solve.
• The natural logarithm of \( e^x \) is simply \( x \), due to the inverse nature of logarithms and exponents.
• Knowing how to apply \( \ln \) helps simplify equations significantly.

For example, in the equation \( e^{1-4x} = 2 \), applying \( \ln \) to both sides gives us \( \ln(e^{1-4x}) = \ln(2) \). By doing this, we are directly using the logarithm property that allows us to extract the exponent easily, transforming it to \( 1 - 4x = \ln(2) \).
Natural logarithms thus make manipulating exponential equations more straightforward.

Solving Equations

Solving equations is a fundamental operation in algebra, and it involves finding the value(s) of the variable(s) that satisfy the equation. When dealing with an exponential equation such as \( e^{1-4x} = 2 \), the goal is to isolate the variable, \( x \), to find its value.
The process often involves several steps:

• Identify the equation and what needs to be solved.
• Apply an appropriate mathematical operation to simplify it. For exponential equations, this often includes taking logarithms, especially natural logarithms.
• Isolate the term containing the variable.
• Perform algebraic steps to solve for the variable.

In solving \( e^{1-4x} = 2 \), after taking the natural logarithm, we get \( 1 - 4x = \ln(2) \). From here, subtracting 1 from both sides isolates the term with \( x \), leading us to \( -4x = \ln(2) - 1 \).
Finally, by dividing by \(-4\), you solve directly for \( x \). Each of these steps simplifies the equation until the variable is isolated, allowing us to find its value numerically.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations or expressions to facilitate solving or understanding. It is a key skill in solving mathematical problems, especially equations that involve algebraic expressions.
When handling an equation such as \( e^{1-4x} = 2 \), algebraic manipulation is vital once you've utilized the natural logarithm. Here is how the manipulation progresses:

• Begin with the simplified equation after applying \( \ln \): \( 1 - 4x = \ln(2) \).
• Subtract 1 from both sides to begin isolating \( x \): \( -4x = \ln(2) - 1 \).
• Divide by \(-4\) to finally isolate \( x \): \( x = \frac{\ln(2) - 1}{-4} \).

These steps are all about balancing the equation by performing identical operations on both sides, eventually leading to a solution.
Each manipulation step aims to reverse the operations affecting \( x \), allowing you to calculate its precise value as shown, \( x \approx 0.0767 \). Understanding these processes and practicing is essential for confidently solving similar algebraic problems.