Browse Questions

If $1$, $log_{b}^{a}$, $log_{c}^{b}$, $-15\;log_{a}^{c}$ are in AP, then

$(a)\;a=c^3\qquad(b)\;a=\frac{1}{b^2}\qquad(c)\;b=\frac{1}{c^2}\qquad(d)\;None\;of\;these$

Answer : (a) $\;a=c^3$
Explanation : Let d be the common difference .
$\;log_{b}^{a}=1+d\quad\;=>\;a=b^{1+d}$
$\;log_{c}^{b}=1+2d\quad\;=>\;b=c^{1+2d}$
$\;-15\;log_{a}^{c}=1+3d\quad\;=>\;c=a^{\frac{-(1+3d)}{15}}$
$a=b^{1+d}=c^{(1+2d)(1+3d)}$
$=a^{\frac{-(1+d)(1+2d)(1+3d)}{15}}$
$(1+d)(1+2d)(1+3d)=-15$
$d=-2$
$a=b^{-1}=c^{(-3)(-1)}=c^{3}\;.$