Single correct option
If log₂x + log₄x + log₁₆x = 21/4, what is the value of log₈(x²√x)?
- A. 2
- B. 5/2
- C. 3
- D. 7/2
Show answer and solution
Answer: (B) 5/2
Convert everything to base 2: log₄x = (1/2)log₂x and log₁₆x = (1/4)log₂x.
(1 + 1/2 + 1/4) log₂x = 21/4 → (7/4) log₂x = 21/4 → log₂x = 3 → x = 8.
x²√x = x^(5/2), so log₈(x^(5/2)) = (5/2) log₈8 = 5/2.
Tip: log_(bⁿ) x = (1/n) log_b x — bring all logs to the smallest common base first.
(1 + 1/2 + 1/4) log₂x = 21/4 → (7/4) log₂x = 21/4 → log₂x = 3 → x = 8.
x²√x = x^(5/2), so log₈(x^(5/2)) = (5/2) log₈8 = 5/2.
Tip: log_(bⁿ) x = (1/n) log_b x — bring all logs to the smallest common base first.
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