Question

Find the solution of the equation , log 𝑥 − log 𝑥 − 1
2 = 2 log( 𝑥 − 1)

Answer 1

Make sure you write the question well.

The solution to what you wrote is:

log x - log x -12 = 2 log ( x - 1 )

- 12 = 2 log ( x - 1 )

Divide both sides by 2

- 6 = log ( x - 1 )

log ( x - 1 ) = - 6

If your logarithm is the logarithm with base 10:

10^ [ log ( x - 1 ) ] = 10^ - 6

x - 1 = 0.000001

Add 1 to both sides

x = 1.000001

Answer 2

Since the equation makes no sense as written,
maybe you meant
log 𝑥 − (log 𝑥 − 12) = 2 log( 𝑥 − 1)
x = 1 + 10^6

Or maybe you meant
log x − log (x − 1)/2 = 2 log( x − 1)
Probably not, since that is equivalent to
(x-1)^3 = 2x

oh, well...