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Your starting equation is
= . (1)
One solution is x= 1 (obvious).
Indeed, left side is = = 1.
Right side is equal to 1 due to the same reason.
Now I will assume that x =/= 1 and will look for other solutions.
Take logarithm base 10 of both sides of equation (1). You will get
= .
Divide both sides by log(x) (we can do it safely, since we consider x =/= 1.)
You will get
= ,
or, equivalently,
= .
It implies
= x
and after squaring both sides,
4x = .
It implies
= 0
x*(4-x) = 0,
x = 0 or x= 4.
You can check that the root x= 0 works in the original equation, since then each side is equal to 1.
So, the remaining solutions to the problem are x= 0 and x= 4.
ANSWER. The given equation has three solutions x= 0, x= 1, and x= 4.
Solved.
To see that x= 0 is the solution, too, look at this plot below.
Plots y = (red) and y = (green)