Problem 31 - College Algebra Essentials

Question

Evaluate each expression without using a calculator. $$\log _{2} \frac{1}{\sqrt{2}}$$

Step-by-Step Solution

Answer

-1/2

Step 1: Break down the expression

The given expression is \(\log _{2} \frac{1}{\sqrt{2}}\). The fraction in the argument of the logarithm can be expressed as a power, exactly \(\frac{1}{\sqrt{2}} = 2^{-1/2}\). Thus, the initial expression can be rewritten as \(\log _{2} 2^{-1/2}\).

Step 2: Apply properties of logarithms

Recall the property of logarithms: The logarithm of a number raised to an exponent is equal to that exponent times the logarithm of the number i.e, \(\log_b a^n = n \cdot \log_b a\). Applying this rule to \(\log _{2} 2^{-1/2}\), we get \(-1/2 \cdot \log _{2} 2\).

Step 3: Simplify the expression

\(\log_b b\) is always equal to 1, for any valid 'b'. Thus, \(\log _{2} 2 = 1\). Substitute this into \(-1/2 \cdot \log _{2} 2\), finally getting -1/2 as the solution of the given expression.