To determine the base [tex]\( b \)[/tex] of the logarithmic function [tex]\( y = \log_b x \)[/tex] from the given tables, follow these steps:
1. Understanding the Function:
- The logarithmic function is given as [tex]\( y = \log_b x \)[/tex]. This means that [tex]\( b^y = x \)[/tex]. Our goal is to find the base [tex]\( b \)[/tex].
2. Using a Known Point:
- Take the point [tex]\((x, y) = (2, 1)\)[/tex] from the table.
- According to the equation [tex]\( y = \log_b x \)[/tex], substituting [tex]\( x = 2 \)[/tex] and [tex]\( y = 1 \)[/tex], we get:
[tex]\[
\log_b 2 = 1
\][/tex]
- This implies that [tex]\( b^1 = 2 \)[/tex], or simply [tex]\( b = 2 \)[/tex].
3. Verification with Another Point:
- Let's verify this base [tex]\( b \)[/tex] using another point. Consider [tex]\((x, y) = \left(\frac{1}{2}, -1\right)\)[/tex].
- Substituting these values into the equation [tex]\( y = \log_b x \)[/tex] gives:
[tex]\[
\log_b \left(\frac{1}{2}\right) = -1
\][/tex]
- This implies that [tex]\( b^{-1} = \frac{1}{2} \)[/tex], or equivalently [tex]\( \frac{1}{b} = \frac{1}{2} \)[/tex].
- Solving this, we get [tex]\( b = 2 \)[/tex].
4. Final Verification and Decision:
- To double-check the correctness, consider that all pairs [tex]\((x, y)\)[/tex] must satisfy [tex]\( y = \log_b x \)[/tex] using [tex]\( b = 2 \)[/tex]. Verifying for a few more points from the table will confirm our result:
- For [tex]\((x, y) = \left(\frac{1}{4}, -2\right)\)[/tex]:
[tex]\[
\log_2 \left(\frac{1}{4}\right) = -2 \implies 2^{-2} = \frac{1}{4}
\][/tex]
- For [tex]\((x, y) = (1, 0)\)[/tex]:
[tex]\[
\log_2 (1) = 0 \implies 2^0 = 1
\][/tex]
Since these points verify correctly using [tex]\( b = 2 \)[/tex], the base [tex]\( b \)[/tex] of the logarithmic function [tex]\( y = \log_b x \)[/tex] is indeed 2.
Thus, the derived base [tex]\( b \)[/tex] for the function [tex]\( y = \log_b x \)[/tex] that fits the given values is:
[tex]\[
\boxed{2}
\][/tex]
