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Laws of logs tells us that ln(a) - ln(b) = ln(a/b)

Therefore ln(10)-ln(5) = ln(10/5) = ln(2)

x² -4=0 becomes (x+2)(x-2)=0

Either one of the two brackets has the value of 0.

If the first bracket is zero, then x+2=0, therefore x=-2

If the second bracket is zero, then...

when differentiating, we multiply the power by the coefficient and then take the power away by 1 so : y= 4x^3 + 3x^2 -5x +1 then 4 x 3= 12 and 3-1 = 2 meaning the solution is 12x^2. 2 x 3 = 6 and 2-1 = 1 ...

Taking the log of both sides we get 5x * ln2 = (2x+1) * ln3.
Taking everything that contains x to the left side: x * (5ln2 - 2ln3) = ln3.
Therefore x=ln3/(5ln2 - 2ln3)
...

divide fraction by 2

3x+2 / x-1  + 3 = 4

multiply by x-1

3x + 2 + 3x - 3 = 4x - 4

6x -1 = 4x -4

2x = -3

x= -3/2