Solve: (1/5)log₈1024 - 2log₈(1/2) Logarithmic Expression

To solve this problem, we'll begin by simplifying the given expression using logarithmic rules:

• Step 1: Simplify the first term $\frac{1}{5}\log_8 1024$:
  Since $1024 = 2^{10}$, we can rewrite this as $\log_8 (2^{10}) = 10 \log_8 2$. Thus, $\frac{1}{5} \log_8 1024 = \frac{1}{5} (10 \log_8 2) = 2 \log_8 2$.
• Step 2: Simplify the second term $2 \log_8 \frac{1}{2}$:
  Note $\frac{1}{2} = 2^{-1}$. Therefore, $\log_8 \left(\frac{1}{2}\right) = \log_8(2^{-1}) = -1 \log_8 2$. Thus, $2 \log_8 \left(\frac{1}{2}\right) = 2(-\log_8 2) = -2 \log_8 2$.
• Step 3: Subtract the expressions:
  Combine the terms using the difference rule:
  $2 \log_8 2 - (-2 \log_8 2) = 2 \log_8 2 + 2 \log_8 2 = 4 \log_8 2$.
• Step 4: Simplify further:
  Since $4 = 2^2$, we can express this as $\log_8 (2^4) = \log_8 (16)$.

Therefore, the simplified form of the expression is $\log_8 16$.

The correct choice is thus $\log_8 16$, matching with choice (1).