Question

If ${10}^{{log}_a\left({log}_b\right({log}_{c}x\left)\right)}=1$ and ${10}^{\left({log}_b\right({log}_c\left({log}_{a}x\right)\left)\right)}=1$, then , a is equal to

• A. a/b
• B. $c^{a/b}$
• C. ab
• D. $c^{b/c}$

Answer 1

The correct option is D $c^{b/c}$

Given,

${10}^{{\log }_a\left({\log }_b\left({\log }_b\left(x\right)\right)\right)}=1$

$\Rightarrow {10}^{{\log }_a\left({\log }_b\left({\log }_b\left(x\right)\right)\right)}={10}^0$

$\Rightarrow {\log }_a\left({\log }_b\left({\log }_b\left(x\right)\right)\right)=0$

$If{\log }_a\left(b\right)=cthenb=a^c$

${\log }_a\left({\log }_b\left({\log }_c\left(x\right)\right)\right)=0\Rightarrow {\log }_b\left({\log }_c\left(x\right)\right)=a^0$

${\log }_b\left({\log }_c\left(x\right)\right)=a^0\Rightarrow {\log }_c\left(x\right)=b^{a^0}$

${\log }_c\left(x\right)=b^{a^0}\Rightarrow x=c^{b^{a^0}}$

$x=c^b$................(1)

${10}^{{\log }_b\left({\log }_c\left({\log }_a\left(x\right)\right)\right)}=1$

$\Rightarrow {10}^{{\log }_b\left({\log }_c\left({\log }_a\left(x\right)\right)\right)}={10}^0$

$\Rightarrow {\log }_b\left({\log }_c\left({\log }_a\left(x\right)\right)\right)=0$

$\Rightarrow {\log }_b\left({\log }_c\left({\log }_a\left(x\right)\right)\right)=0\Rightarrow {\log }_c\left({\log }_a\left(x\right)\right)=b^0$

${\log }_c\left({\log }_a\left(x\right)\right)=b^0\Rightarrow {\log }_a\left(x\right)=c^{b^0}$

${\log }_a\left(x\right)=c^{b^0}\Rightarrow x=a^{c^{b^0}}$

$x=a^c$.............(2)

Comparing (1) and (2), we get,

$c^b=a^c$

raising the power $\frac{1}{c}$ on both sides, we get,

$c^{\frac{b}{c}}=a^{\frac{c}{c}}$

$\Rightarrow c^{\frac{b}{c}}=a$