Problem 54 - A Graphical Approach to College Algebra

Question

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln (5+4 x)-\ln (3+x)-\ln 3=0$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 4 \).

Step 1: Utilize Logarithmic Properties

Recognize that the equation \( \ln(5 + 4x) - \ln(3 + x) - \ln 3 = 0 \) can be simplified using the properties of logarithms. Specifically, recall that \( \ln a - \ln b = \ln \left(\frac{a}{b}\right) \). Therefore, rewrite the given equation as \( \ln \left( \frac{5 + 4x}{3 + x} \right) - \ln 3 = 0 \).

Step 2: Combine Logarithm Terms

Further apply the logarithmic property \( \ln a - \ln b = \ln \left(\frac{a}{b}\right) \) to get the equation \( \ln \left( \frac{5 + 4x}{3(3 + x)} \right) = 0 \). This simplification is crucial to solving the equation in a straightforward manner.

Step 3: Exponentiate Both Sides

To eliminate the logarithm, exponentiate both sides of the equation using \( e \). This means we take \( e^{\ln \left( \frac{5 + 4x}{3(3 + x)} \right)} = e^0 \). The left-hand side simplifies to \( \frac{5 + 4x}{3(3 + x)} = 1 \), because \( e^0 = 1 \).

Step 4: Solve the Resulting Algebraic Equation

Since \( \frac{5 + 4x}{3(3 + x)} = 1 \), this implies that \( 5 + 4x = 3(3 + x) \). Distribute the 3 on the right to get \( 5 + 4x = 9 + 3x \).

Step 5: Isolate the Variable \( x \)

Rearrange the equation from Step 4 to isolate \( x \). Subtract \( 3x \) from both sides to get \( 4x - 3x + 5 = 9 \), which simplifies to \( x + 5 = 9 \). Then, subtract 5 from both sides to solve for \( x \), giving \( x = 4 \).

Step 6: Verify the Solution

To ensure that \( x = 4 \) is a valid solution, substitute it back into the original logarithmic expressions. Calculate \( \ln(5 + 4\cdot4) - \ln(3 + 4) - \ln 3 = \ln 21 - \ln 7 - \ln 3 \). This simplifies to \( \ln \left( \frac{21}{7} \right) - \ln 3 = \ln 3 - \ln 3 = 0 \). The simplified expression satisfies the original equation, confirming that \( x = 4 \) is correct.

Key Concepts

Logarithmic Properties

To solve logarithmic equations, it is important to understand logarithmic properties. These properties allow you to simplify and manipulate logarithmic expressions.
One key property is:

• The difference of two logarithms, such as \( \ln a - \ln b \), can be rewritten as a single logarithm: \( \ln \left(\frac{a}{b}\right) \). This helps to condense expressions, making them easier to solve.

In our example, we applied this property to simplify the equation \( \ln(5 + 4x) - \ln(3 + x) - \ln 3 = 0 \). By recognizing this property, the equation becomes \( \ln \left( \frac{5 + 4x}{3 + x} \right) - \ln 3 = 0 \), which further simplifies to \( \ln \left( \frac{5 + 4x}{3(3 + x)} \right) = 0 \).
Mastering these properties is crucial for efficiently solving logarithmic equations.

Algebraic Equations

After simplifying a logarithmic equation using properties, solve the resulting algebraic equation. In our scenario, the simplified form \( \frac{5 + 4x}{3(3 + x)} = 1 \) results in the algebraic equation \( 5 + 4x = 3(3 + x) \).

• Algebraic equations often involve balancing both sides using operations like addition, subtraction, multiplication, or division.

Here, distributing the 3 over the terms on the right-hand side gives us \( 9 + 3x \). As a result, the equation \( 5 + 4x = 9 + 3x \) is formed. The steps above are fundamental when dealing with algebraic manipulations.

Exponentiation

Exponentiation refers to raising a number to the power of an exponent. In logarithmic problems, exponentiation is often used to cancel out a logarithm. This is done because the natural logarithm, \( \ln \), is the inverse of exponentiation with the base \( e \).
To solve the logarithmic equation \( \ln \left( \frac{5 + 4x}{3(3 + x)} \right) = 0 \), you exponentiate both sides using base \( e \). This process involves changing \( e^{\ln \left( \frac{5 + 4x}{3(3 + x)} \right)} = e^0 \) into \( \frac{5 + 4x}{3(3 + x)} = 1 \).
Since \( e^0 \) equals 1, exponentiation removed the logarithm and simplified the equation, providing a clear path towards solving it.

Variable Isolation

Isolating the variable is the process of manipulating a math equation to get a particular variable by itself on one side of the equation. It is a vital step in solving algebraic equations.
Once the equation is reduced to a simple form, such as \( 5 + 4x = 9 + 3x \), follow these steps:

• Subtract \( 3x \) from both sides to simplify: \( 4x - 3x + 5 = 9 \), resulting in \( x + 5 = 9 \).
• Finally, subtract 5 from both sides to isolate \( x \), giving \( x = 4 \).

By isolating \( x \), the solution becomes clear and can be confirmed by substituting back into the original equation to verify correctness.