Problem 38 - A Graphical Approach to College Algebra

Question

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln (1-x)=\frac{1}{2}$$

Step-by-Step Solution

Answer

\( x = 1 - e^{\frac{1}{2}} \).

Step 1: Understand the Equation

The given equation is \( \ln(1-x) = \frac{1}{2} \). This is a logarithmic equation involving the natural log function, \( \ln \), which denotes a logarithm with base \( e \). Our goal is to solve for \( x \).

Step 2: Exponentiate Both Sides

To eliminate the natural logarithm, raise both sides of the equation as a power of \( e \), which is the base of the natural logarithm:\[e^{\ln(1-x)} = e^{\frac{1}{2}}\]This step simplifies to:\[1-x = e^{\frac{1}{2}}\] because \( e^{\ln(a)} = a \).

Step 3: Solve for x

Now, solve for \( x \) by isolating it on one side of the equation:\[1 - x = e^{\frac{1}{2}}\]Subtract \( e^{\frac{1}{2}} \) from both sides:\[1 = x + e^{\frac{1}{2}}\]Subtract \( e^{\frac{1}{2}} \) from 1:\[x = 1 - e^{\frac{1}{2}}\].

Step 4: Verify Using Calculator

Calculate \( e^{\frac{1}{2}} \) using a calculator, which is approximately 1.6487. Substitute this back into \( x = 1 - e^{\frac{1}{2}} \) to get \( x \approx 1 - 1.6487 = -0.6487 \), verifying our solution is consistent.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithmic function with the base \( e \), where \( e \) is approximately equal to 2.71828. It's known as the "natural" logarithm because of its widespread use in mathematics, especially in calculus and real analysis. Understanding \( \ln \) is crucial in solving exponential and logarithmic equations.
For any positive number \( a \), \( \ln(a) \) tells us the power to which \( e \) must be raised to get \( a \). So if \( \ln(1-x) = \frac{1}{2} \), it means \( e \) raised to the power of \( 0.5 \) equals \( 1-x \). This transformation is a key step in many calculus problems.
It's important to remember:

• The natural logarithm is only defined for positive numbers.
• Common natural logarithm values include \( \ln(1) = 0 \) and \( \ln(e) = 1 \).
• When solving equations involving \( \ln \), removing the logarithm involves exponentiation.

Exponentiation

Exponentiation is the process of raising a number, known as the base, to a power. In the context of logarithmic equations, it's often used to eliminate the logarithm.
For the equation \( \ln(1-x) = \frac{1}{2} \), we can exponentiate both sides with \( e \), giving us \( e^{\ln(1-x)} = e^{\frac{1}{2}} \). This step is crucial because it uses the property \( e^{\ln(a)} = a \), simplifying our equation to \( 1-x = e^{\frac{1}{2}} \).
Here's why exponentiation is helpful when solving equations:

• It allows us to "undo" the logarithm, revealing the variable hidden inside.
• Exponentiation keeps the equation balanced as long as it's applied symmetrically.
• It's a common method used in both theoretical and practical mathematics to simplify expressions.

Isolating Variables

Isolating the variable is a fundamental technique used to solve equations. The goal is to have the variable of interest alone on one side of the equation. In the context of the problem, once we simplified the equation to \( 1 - x = e^{\frac{1}{2}} \), we need to solve for \( x \).
To isolate \( x \), we perform the following steps:

• Rewrite \( 1 - x = e^{\frac{1}{2}} \) as \( 1 = x + e^{\frac{1}{2}} \).
• Subtract \( e^{\frac{1}{2}} \) from both sides to isolate \( x \): \( x = 1 - e^{\frac{1}{2}} \).

Key processes to remember when isolating variables:

• Perform the same operation on both sides of the equation to keep it balanced.
• Simplify each step whenever possible to make the calculations more straightforward.
• Verification via calculator or back-substitution can confirm the correctness of the isolated solution.