Question
Question: Solve the following equation- \({\log ^2}x - 3\log x = \log \left( {{x^2}} \right) - 4\)...
Solve the following equation-
log2x−3logx=log(x2)−4
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:
log2x−3logx=log(x2)−4 = log2x−3logx=2log(x)−4
(since log x2=2logx)
Let log x = t
⇒t2−3t−2t+4=0 ⇒t2−5t+4=0 ⇒t2−4t−t+4=0 ⇒t(t−4)−1(t−4)=0 ⇒t=4 or t=1
Thus, log x = 4 we get
⇒x=104 and
Log x = 1
⇒x=10
Note: Remember to recall the basic logarithmic properties to solve such questions. Note that many students confuse log2x with logx2 but both are very different quantities as logx2 = 2 log x. Don't forget to rewrite the value of log x as t at the final end of the question.