Question

Solve (log x)^2 = log x

Also, when will their be a possibility that an equation will have an extraneous solution? Like if you got two solutions for a log equation, will a negative solution for x automatically be extraneous?

Answer 1

if log x = 1
then the statement is true
so if we are using base 10 logs
then x = 10

Answer 2

(logx)^2 - logx = 0
logx(logx-1) = 0

logx=0 at x=1
logx=1 at x=e assuming natural logs

negative will indeed be extraneous, since log(x) is not defined for x <= 0.

However, if you have something like log(2x-7) then you have to discard any solution where 2x-7 <= 0, not just x<=0.