- Determine the domain of \(\log_2(x+1) \leq \log_3(x+6)\) to be \(x > -1\).
- Calculate the intersection at \(x=3\) by verifying \(f(3) = \log_2(4) = 2\) and \(g(3) = \log_3(9) = 2\).
- Test points in and out of the interval \((-1, 3]\) to confirm that the inequality holds for \(x \leq 3\) and fails for \(x > 3\).
- Conclude that both Jeff's graphs and his solution set \((-1, 3]\) are correct.
- Final answer: \(\boxed{\text{The graphs are correct, and the solution set is correct.}}\)
Explanation
- Analysis of the Inequality and Domain
To determine if Jeff's solution is correct, we analyze the inequality \(\log_2(x+1) \leq \log_3(x+6)\). First, we identify the domain of the inequality. Since the argument of the logarithm must be positive, we require \(x+1 > 0\) and \(x+6 > 0\). This leads to \(x > -1\) and \(x > -6\), so the domain is \(x > -1\). Next, we find the intersection point of the functions \(f(x) = \log_2(x+1)\) and \(g(x) = \log_3(x+6)\) by solving the equation \(\log_2(x+1) = \log_3(x+6)\). At \(x=3\), we have \(f(3) = \log_2(3+1) = \log_2(4) = 2\) and \(g(3) = \log_3(3+6) = \log_3(9) = 2\). Thus, \(x=3\) is a solution to the equation.
- Evaluating the Solution Set and Graphs
We now test the interval \((-1, 3]\) to see if it satisfies the original inequality. For \(x=0\) (which is in the interval), we have \(f(0) = \log_2(1) = 0\) and \(g(0) = \log_3(6) \approx 1.63\). Since \(0 \leq 1.63\), the inequality holds. For \(x=7\) (which is outside the interval, specifically \(x > 3\)), we have \(f(7) = \log_2(8) = 3\) and \(g(7) = \log_3(13) \approx 2.33\). Since \(3 > 2.33\), the inequality does not hold. Because \(f(x)\) grows faster than \(g(x)\) for \(x > 3\), the solution set must be \((-1, 3]\). Jeff's solution set is indeed correct. Since he drew the graphs to illustrate this, his graphs are also correct.
- Conclusion
We have shown that the intersection point of the two logarithmic functions is at \(x=3\) and the inequality holds precisely for \(x \in (-1, 3]\). Jeff correctly identified this interval. As he drew the graphs to support this solution, we conclude that the graphs are correct and the solution set is correct. The correct option is that the graphs are correct, and the solution set is correct.
Examples
Understanding logarithmic inequalities is essential for comparing growth rates in various fields, such as finance or biology. For instance, if you are comparing two investment strategies where the growth of returns follows different logarithmic curves, finding the intersection point helps you identify the period during which one strategy outperforms the other. Mastering these graphical interpretations allows for better decision-making in long-term planning and analysis.