Question

Solve : $\frac{(0.2{)}^{(x-0.5)}}{\sqrt{5}}=5\times (0.04{)}^{(x-1)}$

Answer 1

$\frac{{\left(0.2\right)}^{(x-0.5)}}{\sqrt{5}}=5\times {\left(0.004\right)}^{(x-1)}\Rightarrow \log \left(\frac{{\left(0.2\right)}^{(x-0.5)}}{\sqrt{5}}\right)=\log (5\times {\left(0.004\right)}^{(x-1)})\Rightarrow \log {\left(0.2\right)}^{(x-0.5)}-\log 5^{\frac{1}{2}}=\log 5+\log {\left(0.2\right)}^{2(x-1)}\Rightarrow (x-0.5)log\left(0.2\right)-\frac{1}{2}\log 5=\log 5+2(x-1)\log \left(0.2\right)\Rightarrow (-x+1.5)\log \left(0.2\right)=\frac{3}{2}\log 5\Rightarrow x=1.5-\frac{\log 5\sqrt{5}}{\log \left(0.2\right)}$