Determine the function value.
-Suppose $f ( x ) = \log _ { a } ( x )$ and $f ( 2 ) = 2$. Find $f \left( \frac { \sqrt { 2 } } { 2 } \right)$
A) 2
B) 1
C) $- 2$
D) $- 1$

Question

Quizplus · Accepted Answer

Given $f(2) = 2$, this means $a^2 = 2$, so $a = \sqrt{2}$. Then, $f\left(\frac{\sqrt{2}}{2}\right) = \log_{\sqrt{2}}\left(\frac{\sqrt{2}}{2}\right) = \log_{\sqrt{2}}\left(\sqrt{2}^{1/2}\right) = 1/2$. Since $\sqrt{2} = a$, we have $a^{1/2} = 2^{1/4}$, which simplifies to $2^{1/2}$. Therefore, $f\left(\frac{\sqrt{2}}{2}\right) = -1$ because we need to correct the initial step interpretation: $f\left(\frac{\sqrt{2}}{2}\right) = \log_{\sqrt{2}}\left(\sqrt{2}^{-1}\right) = -1$, as it represents the inverse operation in the exponent to get the base to the argument's value.

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