Question

How was the LHS turned into the RHS equation? Please help me

Answer 1

`\begin{array}{llll} \textit{logarithm of factors} \\\\ \log_a(xy)\implies \log_a(x)+\log_a(y) \end{array} ~\hspace{4em} \begin{array}{llll} \textit{Logarithm of rationals} \\\\ \log_a\left( (x)/(y)\right)\implies \log_a(x)-\log_a(y) \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ t\leqslant \log_(10)\left( \cfrac{5}{3* 10^(-2)} \right) \hspace{5em}t\leqslant 2+\log_(10)\left( \cfrac{5}{3} \right) \\\\[-0.35em] ~\dotfill`

`\log_(10)\left( \cfrac{5}{3* 10^(-2)} \right)\implies \log_(10)\left( \cfrac{5* 10^2}{3} \right)\implies \log_(10)(5* 10^2)-\log_(10)(3) \\\\\\ ( ~~ \log_(10)(5)+\log_(10)(10^2) ~~) ~~ -\log_(10)(3)\implies \log_(10)(5)+2\log_(10)(10)-\log_(10)(3) \\\\\\ \log_(10)(5)+2(1)-\log_(10)(3)\implies 2+\log_(10)(5)-\log_(10)(3)\implies 2+\log_(10)\left( \cfrac{5}{3} \right)`