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Question

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

Solve the following equation-
log2x3logx=log(x2)4{\log ^2}x - 3\log x = \log \left( {{x^2}} \right) - 4

Explanation

Solution

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:
log2x3logx=log(x2)4{\log ^2}x - 3\log x = \log \left( {{x^2}} \right) - 4 = log2x3logx=2log(x)4{\log ^2}x - 3\log x = 2\log \left( x \right) - 4
(since log x2=2logx)\left( {{\text{since log }}{{\text{x}}^2} = 2\log x} \right)
Let log x = t
t23t2t+4=0 t25t+4=0 t24tt+4=0 t(t4)1(t4)=0 t=4 or t=1  \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
x=104\Rightarrow x = {10^4} and
Log x = 1
x=10\Rightarrow x = 10

Note: Remember to recall the basic logarithmic properties to solve such questions. Note that many students confuse log2x{\log ^2}x with logx2\log {x^2} but both are very different quantities as logx2\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.