Question

Evaluate the expression without using a calculator.\(\log _{2} \frac{1}{16}\)

Short answer

-4

Step-by-step solution

- Relationship between log base and number

The first step would be to express 1/16 in terms of 2. We can easily say that \( \frac{1}{16} = 2^{-4} \) as \( 2^4 = 16 \) and 1 divided by 16 is the same as 2 raised to the power of -4.

- Plugging into the logarithm

Now we plug this into the logarithm which now gets simplified as: \( \log _{2} 2^{-4} \)

- Logarithmic rule

We know that the base 2 logarithm of 2 to any power is just the power itself. So, the base 2 logarithm of \( 2^{-4} \) is \( -4 \)