Problem 45 - Precalculus Mathematics for Calculus

Question

Solve the logarithmic equation for \(x .\) $$\log _{2} 3+\log _{2} x=\log _{2} 5+\log _{2}(x-2)$$

Step-by-Step Solution

Answer

The solution is \( x = 5 \).

Step 1: Apply the Properties of Logarithms

Use the property that allows us to combine sums of logarithms into products: \( \log_b A + \log_b B = \log_b (AB) \). Applying this to both sides of the equation, we get: \( \log_2(3x) = \log_2(5(x-2)) \).

Step 2: Remove the Logarithms

Since both sides of the equation have \( \log_2 \), we can set the arguments of the logarithms equal to each other: \( 3x = 5(x - 2) \).

Step 3: Solve the Linear Equation

Expand the equation from Step 2: \( 3x = 5x - 10 \). Rearrange it by subtracting \( 5x \) from both sides: \( 3x - 5x = -10 \), which simplifies to \( -2x = -10 \).

Step 4: Isolate and Solve for x

Divide both sides by \(-2\): \( x = \frac{-10}{-2} \). Simplify to find \( x = 5 \).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is essential in manipulating and solving logarithmic equations. These properties help simplify complex log expressions, and are useful for transforming a sum or difference of logarithms into a single logarithm. Consider the property:

• Product Property: This states that the sum of two logarithms with the same base can be rewritten as the logarithm of the product. Mathematically, we express this as \( \log_b A + \log_b B = \log_b (AB) \). For the equation \( \log_2 3 + \log_2 x = \log_2 5 + \log_2(x - 2) \), we can combine terms to get \( \log_2(3x) = \log_2(5(x-2)) \).

Remember, other logarithm properties can involve differences (quotient rule), powers (power rule), or changing bases (change of base formula), each simplifying different types of logarithmic problems. These transformations are extremely useful in aligning the terms of an equation to isolate and solve for variables.

Solving Linear Equations

Once logarithms are eliminated, solving the resulting equation often involves linear techniques. Here, you'll find that basic algebra rules are applied to isolate the desired variable. After using the properties of logarithms, we get a simple linear equation: \( 3x = 5(x - 2) \). To solve linear equations, follow these steps:

• Expand and Simplify: Distribute the multiplication on the right side to obtain \( 3x = 5x - 10 \).
• Isolate the Variable: Rearrange the equation to gather all terms involving \( x \) on one side. We subtract \( 5x \) from both sides, resulting in \( 3x - 5x = -10 \).
• Final Simplification: Simplify to find \( -2x = -10 \).
• Divide to Solve: Divide each side by \(-2\) to get \( x = 5 \).

These steps underscore basic algebraic techniques that are indispensable for solving equations and verifying the solution is consistent with the original problem statement.

Logarithm Rules

Logarithm rules are the backbone of solving equations involving logarithms. These rules allow for transforming and manipulating logarithmic expressions to solve for unknowns. Here’s a quick recap of important log rules we use in problem-solving:

• Product Rule: \( \log_b A + \log_b B = \log_b (AB) \), used to combine logs.
• Quotient Rule: \( \log_b A - \log_b B = \log_b \left(\frac{A}{B}\right) \), used to handle differences.
• Power Rule: \( \log_b (A^c) = c \cdot \log_b A \), useful if the argument is a power.
• Change of Base Formula: \( \log_b A = \frac{\log_k A}{\log_k b} \), assists in converting bases.

These rules are instrumental when dealing with equations that complicate straightforward algebraic techniques and help convert multi-part logarithmic expressions into a manageable form. They provide a systematic approach to tackle logarithmic equations, provide clarity, and ensure all steps lead to a valid solution.