- Determine the domain constraint \(x > 2\) to ensure the logarithms are defined.
- Combine the logarithmic terms using the property \(\log_b(A) + \log_b(B) = \log_b(AB)\) and rewrite as \(x^2 - 2x - 8 = 0\).
- Factor the quadratic equation \((x - 4)(x + 2) = 0\) to find roots \(x = 4\) and \(x = -2\).
- Exclude the extraneous root \(x = -2\) because it does not satisfy the domain constraint, leaving \(\boxed{4}\).
Explanation
- Identifying the Domain
We start by identifying the domain of the logarithmic equation \(\log_2(x) + \log_2(x - 2) = 3\). For the logarithms to be well-defined, their arguments must be strictly positive. Thus, we require \(x > 0\) and \(x - 2 > 0\). Combining these conditions, we conclude that the variable \(x\) must satisfy \(x > 2\).
- Combining and Converting to Exponential Form
We apply the logarithmic property \(\log_b(A) + \log_b(B) = \log_b(A \cdot B)\) to combine the terms on the left-hand side:
\[\log_2(x(x - 2)) = 3.\]
Using the definition of a logarithm, which states that \(\log_b(Y) = Z\) is equivalent to \(b^Z = Y\), we rewrite the equation in its exponential form:
\[x(x - 2) = 2^3.\]
Since \(2^3 = 8\), we have \(x(x - 2) = 8\).
- Solving the Quadratic Equation
Expanding the left side of the equation \(x(x - 2) = 8\) gives \(x^2 - 2x = 8\). Moving all terms to one side, we obtain the quadratic equation:
\[x^2 - 2x - 8 = 0.\]
To solve for \(x\), we factor the quadratic expression. We look for two numbers that multiply to \(-8\) and add up to \(-2\). These numbers are \(-4\) and \(2\), so we can factor the equation as:
\[(x - 4)(x + 2) = 0.\]
The solutions to this equation are \(x = 4\) and \(x = -2\).
- Verifying the Solutions
We must verify our solutions against the domain constraint \(x > 2\) derived in the first step.
- For \(x = 4\), the condition \(4 > 2\) is satisfied.
- For \(x = -2\), the condition \(-2 > 2\) is not satisfied, so \(x = -2\) is an extraneous solution.
Thus, the only valid solution is \(x = 4\). The final answer is \(\boxed{4}\).
Examples
Logarithmic equations appear frequently in finance when calculating time periods for investments or in science when modeling phenomena such as radioactive decay or the pH level of a solution. For example, if you are modeling the growth of a population or the time it takes for a resource to reach a certain threshold, logarithmic equations allow you to isolate the unknown time variable, enabling you to solve real-world problems involving exponential relationships. Understanding how to solve these equations helps in interpreting data and making informed decisions in various fields.