To determine which table represents the graph of a logarithmic function of the form [tex]\( y = \log_b x \)[/tex] where [tex]\( b > 1 \)[/tex], we need to examine the properties and behavior of logarithmic functions.
A logarithmic function [tex]\( y = \log_b x \)[/tex] has the following characteristics:
1. The function is defined for [tex]\( x > 0 \)[/tex].
2. The function passes through the point [tex]\( (1, 0) \)[/tex] because [tex]\( \log_b(1) = 0 \)[/tex] for any base [tex]\( b \)[/tex].
3. The function increases as [tex]\( x \)[/tex] increases, if [tex]\( b > 1 \)[/tex].
Let's analyze the provided tables:
### First Table
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
\frac{1}{8} & -3 \\
\hline
\frac{1}{4} & -2 \\
\hline
\frac{1}{2} & -1 \\
\hline
1 & 0 \\
\hline
2 & 1 \\
\hline
\end{array}
\][/tex]
In the first table:
- When [tex]\( x = \frac{1}{8} \)[/tex], [tex]\( y = -3 \)[/tex]
- When [tex]\( x = \frac{1}{4} \)[/tex], [tex]\( y = -2 \)[/tex]
- When [tex]\( x = \frac{1}{2} \)[/tex], [tex]\( y = -1 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 0 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 1 \)[/tex]
This table is consistent with the behavior of a logarithmic function, as it:
- Is defined for positive values of [tex]\( x \)[/tex].
- Contains the point [tex]\( (1, 0) \)[/tex].
- Shows [tex]\( y \)[/tex] increasing as [tex]\( x \)[/tex] increases.
### Second Table
The second table is incomplete and does not provide sufficient data to determine whether it represents any logarithmic function. Additionally:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-1.9 & -2096 \\
\hline
-1.75 & -1.262 \\
\hline
- & - \\
\hline
\end{array}
\][/tex]
In the second table:
- It contains negative values for [tex]\( x \)[/tex], which are not within the domain of [tex]\(\log_b x\)[/tex] since [tex]\(x\)[/tex] must be positive.
- The seeming presence of negative [tex]\( x \)[/tex] values and the inconsistency in the [tex]\( y \)[/tex] values do not align with the properties of a logarithmic function with [tex]\( b > 1 \)[/tex].
The first table clearly represents the graph of a logarithmic function in the form [tex]\( y = \log_b x \)[/tex] where [tex]\( b > 1 \)[/tex].
