Problem 11 - Algebra and Trigonometry

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 \( x = 0.0767 \).

Step 1: Understand the Equation

The given equation is \( e^{1-4x} = 2 \). We need to find the value of \( x \) that satisfies this equation.

Step 2: Take the Natural Logarithm

To solve for \( x \), take the natural logarithm of both sides of the equation. This helps in removing the exponential base \( e \).\[ \ln(e^{1-4x}) = \ln(2) \]

Step 3: Simplify Using Logarithmic Identity

Use the property of logarithms that \( \ln(e^y) = y \) to simplify the left side of the equation.\[ 1-4x = \ln(2) \]

Step 4: Isolate the Variable x

Rearrange the equation to solve for \( x \). Subtract 1 from both sides:\[ -4x = \ln(2) - 1 \]Then divide both sides by -4:\[ x = \frac{\ln(2) - 1}{-4} \]

Step 5: Calculate the Solution

Use a calculator to find \( \ln(2) \) and then compute:\[ \ln(2) \approx 0.6931 \]Substitute back:\[ x = \frac{0.6931 - 1}{-4} \]\[ x = \frac{-0.3069}{-4} \]\[ x \approx 0.0767 \]

Key Concepts

Natural Logarithms

Natural logarithms are a fundamental part of many mathematical and real-world applications. They are logarithms with the base of the natural number, denoted as "e," which is approximately equal to 2.71828. The natural logarithm is represented by the notation "\( \ln \)".
Natural logarithms are commonly used in exponential decay and growth processes, such as calculating compound interest or population growth. They help simplify equations involving exponential terms.
Here are some important properties of natural logarithms:

• The natural logarithm of 1 is 0, i.e., \( \ln(1) = 0 \).
• The natural logarithm of "e" itself is 1, i.e., \( \ln(e) = 1 \).
• \( \ln(e^x) = x \), which is the property that simplifies exponential equations in mathematical problems.

By understanding these properties, you can efficiently convert exponential expressions into simpler linear forms that are easier to solve.

Logarithmic Identities

Logarithmic identities are essential tools for simplifying and transforming logarithmic expressions. They allow us to manipulate logarithmic equations to more manageable forms.
There are several key logarithmic identities:

• Product Rule: \( \ln(ab) = \ln(a) + \ln(b) \)
• Quotient Rule: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \)
• Power Rule: \( \ln(a^b) = b \cdot \ln(a) \)
• Logarithm of Exponent: \( \ln(e^x) = x \)

The identity \( \ln(e^y) = y \) is particularly useful when solving exponential equations. It allows us to take logarithms of both sides of an equation and simplify the expressions, reducing the problem to a basic algebraic form.
These identities provide the framework for translating complex exponential equations into formulas that are much easier to work with.

Solving Equations

Solving equations involves finding the unknown variable that makes the equation true. When dealing with exponential equations, the challenge often lies in simplifying the exponential terms.
The process generally includes:

• Taking the natural logarithm of both sides of the equation to remove the exponent. This makes use of the identity \( \ln(e^x) = x \).
• Simplifying the equation using logarithmic identities, if necessary, to get rid of the logarithmic term.
• Rearranging the equation to isolate the variable and solve for it.

In the context of our original exercise, this method helps us find the value of \( x \) that satisfies the equation \( e^{1-4x} = 2 \). By taking the steps of natural logarithms and using logarithmic identities, the problem becomes a more straightforward algebraic equation, which we can then solve efficiently using a calculator to get the numerical result.
Mastery of these techniques in solving equations is essential for success in algebra and higher-level mathematics.