Evaluate the function $f ( x ) = \log _ { 4 } x$ at $x = \frac { 1 } { 16 }$ without using a calculator.
A) $- 1$
B) $- 2$
C) $- 3$
D) $16$
E) $\frac { 1 } { 16 }$

Question

Quizplus · Accepted Answer

We have $f(x)=\log_4x$. 
Substituting $x=\frac{1}{16}$ we get $f\left(\frac{1}{16}ight)=\log_4\frac{1}{16}$.
Using the property $\log_a\frac{1}{x}=-\log_a{x}$ we get $f\left(\frac{1}{16}ight)=-\log_4{16}$.
Since $\log_ab=\frac{1}{\log_ba}$, we have $\log_4{16}=\frac{1}{\log_{16}4}=\frac{1}{\frac{1}{2}}=2.$
Therefore, we get $f\left(\frac{1}{16}ight)=-2$ which is the answer choice $\boxed{	extbf{(B)}}$.

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Evaluate the Function $f ( x ) = \log _ { 4 } x$