I What went wrong in the algebraic steps for ln rules?

[AnotherParadox]

Given

ln(a^b) = b⋅ln(a)

Then

ln(1^x) = x⋅ln(1)

Also

ln(2^x) = x⋅ln(2)

Say

ln(2^x) = ln(1^x)

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?

 

[Ssnow]

Say

$ln(2^x) = ln(1^x)$

this is not true if $x\not= 0$ and, if $x=0$ you can not divide by $x$ here:

Then Also

$x⋅ln(2) = x⋅ln(1)$

Ssnow