Solve the following equation-
${\log ^2}x - 3\log x = \log \left( {{x^2}} \right) - 4$

Hint: In this particular type of question we need to use basic logarithmic properties to simplify the equation and then assume the value of log x as a variable t. Then we need to further solve the equation and finally put the value of t as log x to find the desired answer.

Complete step-by-step solution:
${\log ^2}x - 3\log x = \log \left( {{x^2}} \right) - 4$ = \[{\log ^2}x - 3\log x = 2\log \left( x \right) - 4\]
$\left( {{\text{since log }}{{\text{x}}^2} = 2\log x} \right)$
Let log x = t
$
\Rightarrow {t^2} - 3t - 2t + 4 = 0 \\
\Rightarrow {t^2} - 5t + 4 = 0 \\
\Rightarrow {t^2} - 4t - t + 4 = 0 \\
\Rightarrow t\left( {t - 4} \right) - 1\left( {t - 4} \right) = 0 \\
\Rightarrow t = 4{\text{ or }}t = 1 \\
$
Thus, log x = 4 we get
$ \Rightarrow x = {10^4}$ and
Log x = 1
$ \Rightarrow x = 10$

Note: Remember to recall the basic logarithmic properties to solve such questions. Note that many students confuse ${\log ^2}x$ with $\log {x^2}$ but both are very different quantities as $\log {x^2}$ = 2 log x. Don't forget to rewrite the value of log x as t at the final end of the question.