- Logarithmic functions \(y = \log_b x\) are defined only for \(x > 0\), so Table 2 is invalid because it contains negative \(x\).
- Table 1 contains points that satisfy the function \(y = \log_b x\) with \(b=2\), since \(2^y = x\) holds for all given pairs.
- The base \(b=2\) is greater than 1, which matches the problem requirements.
- Table 1 correctly represents the logarithmic function \(y = \log_2 x\).
- Final Answer: \(\boxed{\text{Table 1}}\)
Explanation
- Analysis of Logarithmic Function Properties
To determine which table represents a logarithmic function of the form \(y = \log_b x\) with \(b > 1\), we must analyze the fundamental properties of such functions. A logarithmic function \(y = \log_b x\) is defined only for \(x > 0\). Furthermore, when the base \(b\) is greater than 1, the function is strictly increasing. Specifically, if \(x=1\), then \(y = \log_b 1 = 0\), and if \(x=b\), then \(y = \log_b b = 1\). Additionally, as \(x\) approaches 0 from the right, the function value \(y\) approaches \(-\infty\).
- Evaluating Table 2
We examine Table 2 first. The provided \(x\) values are \(-1.9\) and \(-1.75\). Since the domain of any logarithmic function \(y = \log_b x\) requires \(x > 0\), Table 2 cannot represent a logarithmic function, as it contains negative values for \(x\). Thus, Table 2 is eliminated.
- Verifying Table 1
Now, we examine Table 1. The points are \((\frac{1}{8}, -3), (\frac{1}{4}, -2), (\frac{1}{2}, -1), (1, 0),\) and \((2, 1)\). All \(x\) values are positive, satisfying the domain requirement. To test if these points fit the form \(y = \log_b x\), we observe that the relation is equivalent to \(b^y = x\). Using the point \((2, 1)\), we have \(b^1 = 2\), which implies \(b = 2\). Since \(2 > 1\), this base satisfies the condition \(b > 1\). We verify this base with the other points:
- For \(x = 1/2\): \(\log_2(1/2) = -1\).
- For \(x = 1/4\): \(\log_2(1/4) = \log_2(2^{-2}) = -2\).
- For \(x = 1/8\): \(\log_2(1/8) = \log_2(2^{-3}) = -3\).
All points in Table 1 satisfy the equation \(y = \log_2 x\). Therefore, Table 1 represents the graph of a logarithmic function with base \(b = 2\).
- Conclusion
Based on the analysis, Table 1 is the only table that corresponds to a valid logarithmic function \(y = \log_b x\) with \(b > 1\), specifically \(y = \log_2 x\). Therefore, Table 1 is the correct choice. The final answer is Table 1.
Examples
Logarithmic functions are used in many real-world applications where values grow exponentially, such as measuring the Richter scale for earthquakes or the pH scale in chemistry. For example, the pH of a solution is defined as \(pH = -\log_{10}[H^+]\), where \([H^+]\) is the concentration of hydrogen ions. If you have a table of ion concentrations and their corresponding pH values, you can use the same method of checking domain and base properties (as we did in this problem) to determine the specific logarithmic relationship and predict the acidity of unknown solutions. This helps scientists and engineers interpret data accurately.