If , , , and , where and , then…

MATHEMATICS · Single correct · A Level · Mathematics · Exponentials and logarithms · Exponentials and logarithms

Problem

\log_a x=\alpha

\log_b x=\beta

\log_c x=\gamma

, and

\log_d x=\delta

, where

x\ne1

and

a,b,c,d>1

, then

\log_{abcd}x

equals (1)

\displaystyle\leq\frac{\alpha+\beta+\gamma+\delta}{16}

(2)

\displaystyle\geq\frac{\alpha+\beta+\gamma+\delta}{16}

(3)

\displaystyle\frac{1}{\alpha^{-1}+\beta^{-1}+\gamma^{-1}+\delta^{-1}}

(4)

\displaystyle\frac{1}{\alpha\beta\gamma\delta}