- Identify the domain constraints: the equation \(\log_2(x) + \log_2(x+7) = 3\) requires \(x > 0\).
- Solve the resulting quadratic equation \(x^2 + 7x - 8 = 0\) to find the candidates \(x = -8\) and \(x = 1\).
- Test the candidates against the domain constraint; only \(x = 1\) satisfies \(x > 0\).
- Conclude that the correct option is \(\boxed{d}\).
Explanation
- Analyzing Domain Constraints
The problem asks us to determine the valid solutions for the logarithmic equation \(\log_2(x) + \log_2(x+7) = 3\). First, we note the domain requirements for the logarithmic functions involved. The expression \(\log_b(u)\) is defined only when the argument \(u\) is strictly greater than zero. Therefore, for \(\log_2(x)\) to be defined, we must have \(x > 0\). Similarly, for \(\log_2(x+7)\) to be defined, we must have \(x+7 > 0\), which simplifies to \(x > -7\). Combining these two conditions, we find that the equation is only defined for \(x > 0\).
- Solving the Algebraic Equation
Following the steps provided in the question, we combine the logarithms using the identity \(\log_b(A) + \log_b(B) = \log_b(AB)\):
\[ \log_2[x(x+7)] = 3 \]
Converting this into exponential form, we get:
\[ x(x+7) = 2^3 \]
\[ x^2 + 7x = 8 \]
Rearranging this into a standard quadratic equation:
\[ x^2 + 7x - 8 = 0 \]
Factoring the quadratic expression:
\[ (x + 8)(x - 1) = 0 \]
This results in two potential solutions: \(x = -8\) and \(x = 1\).
- Evaluating Solutions and Selecting the Correct Option
We now test our potential solutions against the domain constraint \(x > 0\):
- For \(x = -8\): Since \(-8
ot> 0\), this value is outside the domain of the original logarithmic expression. Specifically, \(\log_2(-8)\) is undefined in the set of real numbers.
- For \(x = 1\): Since \(1 > 0\), this value is within the valid domain. We can verify this:
\[ \log_2(1) + \log_2(1+7) = \log_2(1) + \log_2(8) = 0 + 3 = 3 \]
Thus, only \(x = 1\) is a valid solution to the equation. Comparing this with the given choices, option 'd' is the correct answer.
Examples
Logarithmic equations are used in various real-life scenarios, such as calculating the time it takes for an investment to grow under compound interest (using the base-2 logarithm to model binary doubling periods) or in scientific measurements like the Richter scale for earthquakes and the pH scale for chemistry. Ensuring that we account for domain constraints, as done here, is crucial to avoid "extraneous solutions" that do not represent physical reality. For instance, in an investment model, a negative time result would be physically meaningless and must be discarded.