Use base property of logarithm and solve as below,
\(\begin{array}{c}{\log _5}\left( {3x - 1} \right) = {\log _5}\left( {2{x^2}} \right)\\ = 3x - 1\\ = 2{x^2}\\2{x^2} - 3x - 1 = 0\end{array}\)
Now, factor the above equations as,
\(\begin{array}{c}2x\left( {x - 1} \right) - 1\left( {x - 1} \right) = 0\\\left( {x - 1} \right)\left( {2x - 1} \right) = 0\end{array}\)
To find the values of x, use the Zero Product Property,
\(\begin{array}{c}2x - 1 = 0\\x = \frac{1}{2}\end{array}\)
Or,
\(\begin{array}{c}x - 1 = 0\\x = 1\end{array}\)
Hence, the solutions are 1 and \({\frac{1}{2}}\).
\(\begin{array}{c}{\log _5}\left( {3x - 1} \right) = {\log _5}\left( {2{x^2}} \right)\\ = 3x - 1\\ = 2{x^2}\\2{x^2} - 3x - 1 = 0\end{array}\)
Now, factor the above equations as,
\(\begin{array}{c}2x\left( {x - 1} \right) - 1\left( {x - 1} \right) = 0\\\left( {x - 1} \right)\left( {2x - 1} \right) = 0\end{array}\)
To find the values of x, use the Zero Product Property,
\(\begin{array}{c}2x - 1 = 0\\x = \frac{1}{2}\end{array}\)
Or,
\(\begin{array}{c}x - 1 = 0\\x = 1\end{array}\)
Hence, the solutions are 1 and \({\frac{1}{2}}\).