log(4 cube root 4^2) (1/1024) is equal to

Question

log\(_{4\sqrt[3]{4^2}}\) \(\bigg(\frac{1}{1024}\bigg)\) is equal to

(a) – 5 

(b) – 3 

(c) 3 

(d) 5

Answer 1

(b) - 3

Hint. 

Since \({4\sqrt[3]{4^2}}\) = 2².(2⁴)^{\(^{\frac{1}{3}}\)}= \(2^{2+\frac{4}{3}}\) = \(2^{\frac{10}{3}}\) and 

1024 = 2¹⁰, therefore,

log\(_{4.\sqrt[3]{4^2}}\) \(\bigg(\frac{1}{1024}\bigg)\) = log_2^10/3 (2^-10) = \(\frac{-10}{\frac{10}{3}}\) log₂ 2

= – 3 × 1 = – 3. \(\bigg[\)Using log_a x^m = \(\frac{m}{n}\) log_a x\(\bigg].\)