Tricky logarithm problem

Question

I having a problem in this logarithm problem involving modulus- Solve for x

|x-1|^((log(x))^2-2log(x))=|x-1|^3

Bases same so powers equal.

If I take log x as a then I get the following quadratic- A^2-2a-3=0

So x values are 1000 and 0.1. Which is correct. Now there is one more solution to the question which is x=2.

When I subsitute 2 then obviously it is satisfying the equation. But how do we derive it ? Pls help

Answer 1

General solution to the equation $|x-1|^{log(x)^2-2log(x)}=|x-1|^3$:

• $log(x)^2-2log(x)=3$
• $|x-1|=1$
• $[|x-1|=0]\wedge[log(x)^2-2log(x)\neq0]$

Pitfalls to watch for:

• Make sure that the solution doesn't yield $0^0$
• Make sure that the solution doesn't yield $log(0)$