Problem 49 - Algebra and Trigonometry

Question

Solve the logarithmic equation for \(x.\) \(\log _{5}(x+1)-\log _{5}(x-1)=2\)

Step-by-Step Solution

Answer

\(x = \frac{13}{12}\)

Step 1: Use the Quotient Rule for Logarithms

The equation \( \log_{5}(x+1) - \log_{5}(x-1) = 2 \) allows us to use the quotient rule of logarithms. This rule states \( \log_{b}(A) - \log_{b}(B) = \log_{b}\left(\frac{A}{B}\right) \). Apply this rule to get \( \log_{5}\left(\frac{x+1}{x-1}\right) = 2 \).

Step 2: Convert Logarithmic Equation to Exponential Form

The equation \( \log_{5}\left(\frac{x+1}{x-1}\right) = 2 \) can be rewritten in exponential form. This gives \( \frac{x+1}{x-1} = 5^{2} \), which simplifies to \( \frac{x+1}{x-1} = 25 \).

Step 3: Solve the Proportion for \(x\)

Set the equation \( \frac{x+1}{x-1} = 25 \) as a proportion and solve for \( x \). Cross-multiply to get \( x+1 = 25(x-1) \). Then distribute on the right to get \( x+1 = 25x - 25 \).

Step 4: Isolate \(x\)

Rearrange the equation \( x+1 = 25x - 25 \) to isolate \( x \). Subtract \( x \) from each side to get \( 1 = 24x - 25 \), then add 25 to both sides to get \( 26 = 24x \).

Step 5: Divide to Find \(x\)

To solve for \( x \), divide both sides by 24: \( x = \frac{26}{24} \). Simplify to get \( x = \frac{13}{12} \).

Step 6: Verify the Solution

Verify that \( x = \frac{13}{12} \) satisfies the original equation \( \log_{5}(x+1) - \log_{5}(x-1) = 2 \). When substituting back into the equation, both sides should evaluate to 2, confirming the solution is correct.

Key Concepts

Quotient Rule of Logarithms

The quotient rule of logarithms is a powerful tool that simplifies logarithmic expressions involving subtraction. This rule states that for the same base, the difference of two logarithms is equivalent to a single logarithm with a quotient of the arguments. Mathematically, it is expressed as: \[ \log_{b}(A) - \log_{b}(B) = \log_{b}\left(\frac{A}{B}\right) \]In the original exercise, the equation \( \log_{5}(x+1) - \log_{5}(x-1) = 2 \) applied the quotient rule. This combined the two separate logarithms into one:

• \( \log_{5}\left(\frac{x+1}{x-1}\right) = 2 \)

This step significantly simplifies the problem, reducing the complexity of further calculations. Understanding the quotient rule is crucial when dealing with more advanced logarithmic problems, as it helps streamline the equation into a more manageable form.

Exponential Form

Converting a logarithmic equation to its exponential form is essential in solving equations involving logarithms. Remember, a logarithmic expression \( \log_{b}(A) = C \) can be transformed into an exponential form as \( A = b^{C} \). In the given problem, the equation transformed into exponential form is:

• \( \log_{5}\left(\frac{x+1}{x-1}\right) = 2 \) becomes \( \frac{x+1}{x-1} = 5^{2} \)

This conversion is important because it translates the problem from logarithmic language, which can sometimes be unintuitive, into a format that is often easier to manipulate algebraically. By recognizing this equivalence, you can simplify and continue working towards solving for \( x \). It's an accessible way to approach a logarithmic problem using familiar exponential functions.

Proportion

After converting the logarithmic expression to an exponential form, you encounter a proportion, a mathematical equation that states two ratios or fractions are equivalent. In this scenario, \( \frac{x+1}{x-1} = 25 \).To solve this proportion, cross-multiplication is used, which involves multiplying the numerator of one fraction by the denominator of the other and vice-versa. The steps are as follows:

• Cross-multiply: \( x+1 = 25(x-1) \)
• Distribute the 25: \( x+1 = 25x - 25 \)

This prepares the equation for solving by isolating \( x \). Cross-multiplying is a reliable method to solve proportions, allowing for straightforward simplification of the equation. Understanding proportions will enhance your algebra skills, giving you tools to deal with a wide range of mathematical problems involving fractions or ratios.