Problem 54 - A Graphical Approach to Precalculus with Limits

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: Use Logarithm Properties

Use the properties of logarithms to simplify \( \ln (5+4x) - \ln (3+x) - \ln 3 = 0 \). Recall that \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \). Therefore, the equation becomes \( \ln \left( \frac{5+4x}{3+x} \right) - \ln 3 = 0 \).

Step 2: Simplify Logarithmic Expression

Simplify further by using the property \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \). Thus, \( \ln \left( \frac{\frac{5+4x}{3+x}}{3} \right) = 0 \). This simplifies to \( \ln \left( \frac{5+4x}{3(3+x)} \right) = 0 \).

Step 3: Exponential Form

Set the logarithmic expression equal to zero by using the property that \( \ln(a) = 0 \) implies \( a = 1 \). Thus, \( \frac{5+4x}{3(3+x)} = 1 \).

Step 4: Solve the Equation

Solve the equation \( \frac{5+4x}{3(3+x)} = 1 \). Cross-multiply to get \( 5 + 4x = 3(3 + x) \). Simplify the right side to get \( 5 + 4x = 9 + 3x \).

Step 5: Isolate x

Bring like terms together: \( 4x - 3x = 9 - 5 \). Simplify to find \( x = 4 \).

Step 6: Verify the Solution Using a Calculator

Plug the solution back into the original equation to ensure accuracy. Substitute \( x = 4 \) into \( \ln(5+4 \times 4) - \ln(3+4) - \ln 3 \). Calculate each term using a calculator: \( \ln(21) - \ln(7) - \ln 3 = 0 \). Confirm that the terms balance to 0.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms can greatly simplify complex equations. These properties allow us to transform and reduce logarithmic expressions for easier manipulation. Two fundamental properties are pivotal in solving logarithmic equations like

• Difference of Logarithms: This allows you to transform the subtraction of logarithms into the logarithm of a quotient. It is expressed as: \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \).
• Equality to Zero: If \( \ln(a) = 0 \), this means that \( a = 1 \). This property is crucial when setting up equations to easily find the variables.

In the original exercise, we used these properties to simplify the equation \( \ln(5+4x) - \ln(3+x) - \ln 3 = 0 \). Applying the difference property led us to \( \ln \left( \frac{5+4x}{3+x} \right) - \ln 3 = 0 \). Continuing, this became \( \ln \left( \frac{\frac{5+4x}{3+x}}{3} \right) = 0 \). This is a fundamental step in solving any logarithmic equation.

Solving Equations

Solving logarithmic equations usually involves a few strategic transformations and simplifications. Once we have reduced the logarithmic expressions using properties, the next step is to convert them into an algebraic form that can be solved more easily. For the given problem, after simplification, we tackle the equation \( \frac{5+4x}{3(3+x)} = 1 \). This involves the technique of cross-multiplying, which is used to eliminate fractions and solve for the unknown variable.

• Cross-Multiplication: This involves multiplying both sides of the equation to clear the fraction, giving us \( 5 + 4x = 3(3 + x) \).
• Simplification: Distribute and collect like terms to isolate the variable \( x \), simplifying from \( 5 + 4x = 9 + 3x \) to \( 4x - 3x = 9 - 5 \).

This straightforward equation lets us find that \( x = 4 \). Breaking down into these clear steps can help demystify the equation-solving process and ensure accuracy.

Exponential Form

Converting logarithmic equations into their exponential form is often the step that directly reveals the solution. This conversion is built on the basic understanding that logarithms and exponentials are inverse functions. In essence, transforming a logarithm to an exponential form follows from the equation property that if \( \ln(a) = 0 \), then \( a = 1 \), which helps isolate parts of a logarithmic equation. For the given exercise, when you set \( \ln \left( \frac{5+4x}{3(3+x)} \right) = 0 \), this is equivalent to saying \( \frac{5+4x}{3(3+x)} = e^0 = 1 \). From here, we resolve to the equation without logarithms, making it simpler to solve. Each time you convert into over plain arithmetic or algebra, you gain clarity, shedding the complexity introduced by the logarithm.