Problem 23 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$ \ln (-4-x)+\ln 3=\ln (2-x) $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = -7 \).

Step 1: Understand the Logarithmic Properties

The equation given is a logarithmic equation: \( \ln (-4-x) + \ln 3 = \ln (2-x) \). One of the fundamental properties of logarithms is that the sum of logs is the log of the product. Therefore, \( \ln (-4-x) + \ln 3 \) can be rewritten as a single logarithm: \( \ln ((-4-x) \cdot 3) \).

Step 2: Simplify the Equation

Using the property from Step 1, the equation becomes \( \ln (3(-4-x)) = \ln (2-x) \). We can remove the natural log by equating the arguments (since \( \ln a = \ln b \) implies \( a = b \)): \( 3(-4-x) = 2-x \).

Step 3: Expand and Rearrange the Equation

Expand the left side: \( 3(-4-x) = -12 - 3x \). Thus the equation now reads: \( -12 - 3x = 2-x \). Move the terms involving \( x \) to one side and the constant terms to the other side.

Step 4: Solve for x

Add \( 3x \) to both sides of the equation: \( -12 = 2 - x + 3x \), which simplifies to \( -12 = 2 + 2x \). Subtract 2 from both sides to isolate the term involving \( x \): \( -14 = 2x \). Divide both sides by 2 to solve for \( x \): \( x = -7 \).

Step 5: Check the Solution

Substitute \( x = -7 \) back into the original equation to verify it works: the left side becomes \( \ln (-4-(-7)) + \ln 3 = \ln(3) + \ln(3) = 2\ln(3) \). The right side is \( \ln(2-(-7)) = \ln(9) = 2\ln(3) \). Both sides equal, indicating \( x = -7 \) is the correct solution.

Key Concepts

Properties of Logarithms

Logarithms have unique properties that simplify complex expressions, making equations easier to solve. One key property is that the sum of two logarithms, such as \( \ln a + \ln b \), can be expressed as a single logarithm \( \ln(ab) \). This transformation is crucial when working with logarithmic equations because it allows us to consolidate terms.
Another important property is the power rule: \( \ln(a^b) = b\ln(a) \). Although not used in this specific problem, it's helpful for understanding the flexibility of logarithmic expressions. These properties are immensely useful for manipulating equations and finding solutions.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a specific type of logarithm with base \( e \), where \( e \) is approximately 2.718. The natural logarithm is commonly used in calculus and mathematical modeling because of its natural base, which simplifies many equations involving growth and decay.
In the given equation \( \ln(-4-x) + \ln 3 = \ln(2-x) \), understanding \( \ln \) helps us simplify and solve the equation more effectively. Using the properties of \( \ln \), we can remove the logarithms and equate the arguments when they appear correctly on both sides of the equation.

Solving Equations

Solving logarithmic equations often involves isolating the logarithms and using algebraic manipulation. In the equation \( \ln(-4-x) + \ln 3 = \ln(2-x) \), we first use the property of logarithms to combine the left side: \( \ln((3)(-4-x)) = \ln(2-x) \).
After simplifying, the equation \( 3(-4-x) = 2-x \) emerges. We can then apply algebra to isolate \( x \): first by distributing and then by rearranging terms to bring terms containing \( x \) to one side and constants to the other. This step-by-step approach ensures clarity and reduces errors during computation.

Verification of Solutions

Verification plays a crucial role in confirming if the obtained solution is correct. After determining that \( x = -7 \), we substitute it back into the original equation.

• Left Side: \( \ln(-4 - (-7)) + \ln 3 = \ln(3) + \ln(3) = 2\ln(3) \).
• Right Side: \( \ln(2 - (-7)) = \ln(9) = 2\ln(3) \).

Both sides equate, proving that our solution \( x = -7 \) satisfies the original logarithmic equation. Verification ensures that no errors occurred during simplification and that the logic applied is sound.