Problem 44 - College Algebra

Question

Solve each logarithmic equation in Exercises \(27-44 .\) Be sure to reject any value of \(x\) that produces the logarithm of a negative number or the logarithm of \(0 .\) $$\log _{2}(x-3)+\log _{2} x-\log _{2}(x+2)=2$$

Step-by-Step Solution

Answer

The valid solution of the given logarithmic equation is \( x = 4 \)

Step 1: Combine logarithm terms using log properties

Making use of the product and quotient logarithm properties, the given logarithmic equation can be combined into one single log term. Apply the formula \( \log_b{a} - \log_b{b} = \log_b{\frac{a}{b}} \) and \( \log_b{a} + \log_b{b} = \log_b{ab} \). The equation becomes \[ \log_2{\left(\frac{(x-3)x}{(x+2)}\right)} = 2 \]

Step 2: Apply definition of logarithm

Applying the definition of a logarithm which is \( \log_b{a} = c \) if \( b^c = a \), the equation can be rewritten as \[ \frac{(x-3)x}{(x+2)} = 2^2 = 4 \]

Step 3: Solve the resulting quadratic equation

Now, a quadratic equation is obtained which needs to be solved for \( x \). Cross multiply to move all terms to one side and factor the resulting quadratic expression. Solving this equation yields two solutions \[ x = 4, -1 \]

Step 4: Check validity of solutions and reject any invalid solutions

Checking the two solutions on the original equation, substitution of \( x = -1 \) into the equation would lead to taking logarithm of a negative number, which is not valid. Therefore, this solution should be rejected. The other solution \( x = 4 \) satisfies the equation and does not lead to taking the logarithm of zero or a negative number, thus it is the valid solution.

Key Concepts

Logarithm Properties

Logarithm properties are essential tools for simplifying complex logarithmic expressions. These properties help us combine or separate logarithmic terms, depending on what the situation requires. Here are two key properties that are often used:

• Product Property: \( \log_b{(a \cdot c)} = \log_b{a} + \log_b{c} \)
• Quotient Property: \( \log_b{\left( \frac{a}{c} \right)} = \log_b{a} - \log_b{c} \)

In our exercise, these properties are applied to transform multiple logarithmic terms into a single logarithmic equation. This process involves combining the terms using the product and quotient properties before solving the equation step-by-step. Thus, these properties are powerful tools in simplifying and resolving logarithmic equations.

Definition of Logarithm

The definition of a logarithm is fundamental to understanding and solving logarithmic equations. Essentially, a logarithm answers the question: "To what power must the base be raised to produce a given number?" Mathematically, this is expressed as:

• \( \log_b{a} = c \) means \( b^c = a \).

In our context, once we combine the logarithmic terms, we find an equation of the form \( \log_b{a} = c \). Applying the definition, we rewrite this in its exponential form, which then becomes a more straightforward equation to solve. This conversion from logarithmic to exponential form is a crucial step in dealing with logarithmic equations.

Quadratic Equation

After applying the definition of logarithm, we often reach a quadratic equation. Quadratic equations have the general form \( ax^2 + bx + c = 0 \), and they can be solved using several methods such as factoring, completing the square, or the quadratic formula.

• Factoring: Rearrange and find numbers that multiply to \( ac \) and add to \( b \).
• Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \)

In the exercise, the resulting quadratic equation after applying the exponential definition is solved by cross-multiplying and simplifying, which leads to our factorable quadratic expression. This enables us to find the potential solutions by finding the roots of the equation.

Validity of Solutions

When solving logarithmic equations, checking the validity of solutions is necessary. Some solutions might not be permissible in the context of logarithms due to the domain restrictions. Logarithms are undefined for negative numbers and zero. Therefore, any solution that leads to such conditions in the original logarithmic equation must be rejected.

• Check each solution by substituting back into the original equation.
• Ensure that the arguments of all logarithms are positive and non-zero.

In our example, we find two potential solutions for the quadratic equation. However, upon substitution, one yields an invalid logarithm (undefined), hence it must be discarded. Only solutions that keep the original equation valid under logarithm rules are acceptable.