Problem 52 - Intermediate Algebra

Question

Solve each equation. $$ \ln \left(x^{2}+4 x\right)=\ln \left(x^{2}+16\right) $$

Step-by-Step Solution

Answer

The solution is \( x = 4 \).

Step 1: Understand The Equation

The equation given is \( \ln(x^2 + 4x) = \ln(x^2 + 16) \). Since the \( \ln \) function is applied to both sides, if the logs are equal, then the expressions inside the logs must also be equal for the equality to hold. Thus, we can set the expressions inside the logarithms equal to each other.

Step 2: Set The Expressions Equal

Since \( \ln(A) = \ln(B) \) implies \( A = B \) (assuming both sides are defined), set the expressions inside the ln equal: \[ x^2 + 4x = x^2 + 16 \] Simplify this equation by eliminating \( x^2 \) from both sides.

Step 3: Simplify and Solve for x

Subtract \( x^2 \) from both sides to simplify the equation: \[ 4x = 16 \] Next, divide each side by 4 to solve for \( x \): \[ x = 4 \].

Step 4: Verify the Solution

Substitute \( x = 4 \) back into the original expressions inside the logarithms to ensure both sides of the original equation are equal:- Left side: \( x^2 + 4x = 4^2 + 4 \times 4 = 16 + 16 = 32 \).- Right side: \( x^2 + 16 = 4^2 + 16 = 16 + 16 = 32 \). Both sides are equal, confirming \( x = 4 \) is correct.

Key Concepts

Properties of Logarithms

Logarithms are powerful tools in mathematics, helping to simplify and solve complex equations. The fundamental property we harness here is that if two logarithms with the same base are equal, their arguments must also be equal, given that they are defined.
This stems from the one-to-one nature of the logarithmic function, meaning that

• if \( \ln(A) = \ln(B) \), then \( A = B \).

Another property of logarithms is that they cannot take negative or undefined values. This restricts our solutions to those numbers that make the argument of a logarithm positive. For example, in our given equation, \( x^2 + 4x \) and \( x^2 + 16 \) must both be greater than zero, ensuring the equation makes sense in real numbers.
Understanding these critical properties allows us to translate logarithmic equations into algebraic ones, simplifying the problem-solving process.

Equation Solving

Solving the equation \( \ln(x^2 + 4x) = \ln(x^2 + 16) \) involves leveraging the property discussed where the arguments inside the logarithms are set equal to one another. By doing this, we bypass the logarithms and focus solely on the algebraic expressions.
Starting with:

• \( x^2 + 4x = x^2 + 16 \)

We first simplify by removing \( x^2 \) from both sides:

• \( 4x = 16 \)

Next, divide by 4 to solve for \( x \):

• \( x = 4 \)

This solution process emphasizes basic algebraic manipulation, showcasing how complex equations can be stripped down to simpler forms for easy handling.

Algebraic Verification

Verification is a vital step in solving equations to ensure that solutions are correct and valid within the context of the original problem. Here, we substitute \( x = 4 \) back into the expressions from the original equation to verify their equality.
The calculations are straightforward:

• For the left side: \( x^2 + 4x = 4^2 + 4 \times 4 = 16 + 16 = 32 \)
• For the right side: \( x^2 + 16 = 4^2 + 16 = 16 + 16 = 32 \)

Since both sides are equal, \( x = 4 \) is confirmed as the correct solution. However, it's crucial also to ensure the argument inside the logarithms remains positive, verifying the solution doesn't breach the domain constraints. This step guarantees that our answer not only solves the equation but also adheres to the principles underpinning the logarithmic function.