Mathematics       

Here x 2 > 0 for all real numbers x. Therefore the equation  x 2 = b only has solutions x when b > 0, that is only when b is non-negative.  Defining
b½ =sqrt(b)
as the nonnegative real solution of  x 2 = b works only  if b is positive. …

 Mathematics       

Now (a x)y = exp(y ln(a x )) =   exp(y x ln(a )) =  a yx = a xy Therefore
(a x)y =  a xy (Exponential of an exponential)
Now a xay =  exp(x ln(a)) · exp(y ln(a) = exp(x ln(a)+y ln(a)) = exp( (x +y )ln(a) ) = a x+y Therefore  …

 Mathematics       

For x = m/n and a > 0,  a x = a m/n = exp( (m/n) ln(a)) = exp( x ln(a)). This suggests putting a x = exp( x ln(a)) for x irrational.  Then
a x = exp( x ln(a)) for all real x for a > …

 Mathematics       

today I was preparing myself for the math exam that I have for the next week
I learned the exponentional exp(x) and logarithms ln(x) functions
Uniqueness (or 1 to 1) Property:
If a > 0, b> 0 and  ln(a) = ln(b) then a = b..
Inversion Properties

ln(exp(x)) = x for all real x
exp(ln(x)) = …