Question

evaluate the logarithm without using a calculator. \\(\log_{2} \frac{1}{16}\\) \\(\log_{2} \frac{1}{16} = \square\\)

Explanation:

Step1: Recall logarithm definition

Let \( \log_{2}\frac{1}{16} = x \). By definition of logarithm, \( 2^{x}=\frac{1}{16} \).

Step2: Express \( \frac{1}{16} \) as power of 2

We know \( \frac{1}{16}=2^{-4} \) since \( 2^{4} = 16 \), so \( \frac{1}{16}=\frac{1}{2^{4}}=2^{-4} \).

Step3: Solve for \( x \)

Now we have \( 2^{x}=2^{-4} \). Since the bases are the same, the exponents must be equal, so \( x = - 4 \).

Answer:

\(-4\)