I Exponential Log Rules Issue?

Given

ln(ab) = b⋅ln(a)

Then

ln(1x) = x⋅ln(1)

Also

ln(2x) = x⋅ln(2)

Say

ln(2x) = ln(1x)

Then Also

x⋅ln(2) = x⋅ln(1)

But, dividing both sides by x

ln(2) ≠ ln(1)

Similarly,

x⋅ln(2) = x⋅ln(1)

Dividing both sides by x and ln(2)

1 ≠ 0

But we know x = 0 as per the original statement.

The question then is which algebraic step(s) was(were) wrong, and why?
 

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