If ${\text{log}}_{\text{1}0}\left({\text{ab}}^{\text{3}}\right)={\text{log}}_{\text{1}0} \left({\text{a}}^{\text{2}}\text{b}\right)=\text{1}$ , then the value of ${\text{log}}_{\text{1}0}\left(\text{ab}\right)$ is equal to

We have ${\text{log}}_{\text{1}0 }\left({\text{ab}}^{\text{3}}\right)=\text{ 1}$           $\Rightarrow$      ${\text{ab}}^{\text{3}} =\text{1}0$        ......... (1)

Also${\text{log}}_{\text{1}0} \left({\text{a}}^{\text{2}}\text{b}\right)=\text{ 1}$        $\Rightarrow$      ${\text{a}}^{\text{2}}\text{b=1}0$         ......... (2)

$\therefore$  From equation (1) and equation (2), we get ${\text{ab}}^{\text{3}} ={\text{ a}}^{\text{2}}\text{b}$

$\Rightarrow$      $\text{ab}\left({\text{b}}^{\text{2}} –\text{ a}\right)=0$  

$\Rightarrow$  ${\text{b}}^{\text{2}} =\text{ a}$ (As a $\ne$  0, b $\ne$  0)

Now on putting  $\text{a }={\text{ b}}^{\text{2}}$ in equation (1), we get ${\text{b}}^{\text{5}} =\text{ 1}0$

$\therefore$ $b={10}^{\frac{1}{5}}$          and $a={10}^{\frac{2}{5}}$

Hence ${\text{log}}_{\text{1}0}\left(\text{ab}\right)$ = ${\text{log}}_{\text{1}0}$ $\left({10}^{\frac{2}{5}}·{10}^{\frac{1}{5}}\right)$   = ${\text{log}}_{\text{1}0}$ $\left({10}^{\frac{3}{5}}\right)$  =  Ans. ]