Question 5 Evaluate the expression: $$ \log _{1 / 2} 64 $$.

Question

UpStudy AI · Accepted Answer

1. We start by letting the logarithmic expression equal $$ x $$:
   $$
   \log_{1/2} 64 = x
   $$

2. By the definition of logarithms, this means:
   $$
   \left(\frac{1}{2}\right)^x = 64
   $$

3. We can rewrite $$ \frac{1}{2} $$ as $$ 2^{-1} $$:
   $$
   \left(2^{-1}\right)^x = 64 \quad \Longrightarrow \quad 2^{-x} = 64
   $$

4. Express $$ 64 $$ as a power of $$ 2 $$, noting that $$ 64 = 2^6 $$:
   $$
   2^{-x} = 2^6
   $$

5. Since the bases are the same, we equate the exponents:
   $$
   -x = 6
   $$

6. Solve for $$ x $$:
   $$
   x = -6
   $$

Thus, the value of $$ \log_{1/2} 64 $$ is $$ -6 $$.
$$ \log_{1/2} 64 = -6 $$

Question 5 Evaluate the expression: \( \log _{1 / 2} 64 \).