Question

If ${\left(150\right)}^{\mathrm{x}}=7$, then $\mathrm{x}$ is equal to :

Answer 1

Solve the given expression for $\mathrm{x}$

Taking logarithm on both sides

$\log {\left(150\right)}^{\mathrm{x}}=\log \left(7\right)$

$\Rightarrow$ $\mathrm{xlog}\left(150\right)=\log \left(7\right)$ $...\left(\log {\left(\mathrm{a}\right)}^{\mathrm{m}}\right)=\mathrm{mlog}\left(\mathrm{a}\right)$

$\Rightarrow$ $\mathrm{x}=\frac{\log \left(7\right)}{\log \left(150\right)}$

$\Rightarrow$ $\mathrm{x}=\frac{\log \left(7\right)}{\log \left(10\times 3\times 5\right)}$

$\Rightarrow$ $\mathrm{x}=\frac{\log \left(7\right)}{\log \left(10\right)+\log \left(3\right)+\log \left(5\right)}$

$\Rightarrow$ $\mathrm{x}=\frac{\log \left(7\right)}{1+\log \left(3\right)+\log \left(5\right)}$ $...\left(\log \left(10\right)=1\right)$

Hence, the value of $\mathrm{x}$ is $\frac{\log \left(7\right)}{1+\log \left(3\right)+\log \left(5\right)}$.