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

b^½ =sqrt(b)

as the nonnegative real solution of  x ² = b works only  if b is positive. This solution is given by a ^½ = exp( ½ln(b)). See above.

Similarly, if n = 2m > 0 is an even, then x ⁿ = x ^2m > 0 for all real numbers x. So   the equation  x ^2m = b only has solutions x when b > 0, that is only when b is non-negative. The foregoing implies defining

b^½m =as the 2m root of (b)

as the nonnegative real solution of  x ^2m = b works only  if b is positive.  This solution is then given by a^1/n = exp( (1/n)ln(b)) where n = 2m.  See above

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 ^xa^y =  exp(x ln(a)) · exp(y ln(a) = exp(x ln(a)+y ln(a)) = exp( (x +y )ln(a) ) = a ^x+y Therefore  has the exponential property

a ^xa^y = a ^x+y for all real numbers x and y when a > 0.

Now  for the natural number e = exp(1) = 2.718281828…  (irrational, deci), the natural logarithm of e,  ln (e) = 1 Therefore

e ^x = exp( x) for all real x when a > 0

as a ^x = exp( x ln(a)). Calculators often have a button marked e ^x for the evaluation of the exponential function exp( x)

Caution: the capital EXP on some calculators will not help you with the calculation of exp(x). Use the  button marked e ^x instead.

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 > 0

and not only for rational numbers. From this definition,  ln a ^x =  x ln(a).  Therefore log_a(a ^x) = x  because  log_a(x) = ln(x)/ln(a).

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)) = x if x > 0

For each real number a,   x = exp(a) is the unique solution of  a =  ln(x).  Solving the latter equation  is one way to define or compute exp(a).

Fundamental property of logarithms

ln(ab) = ln(b) +ln(a)    (proof available in calculus)

Fundamental property of exponentials:

exp(x₁) · exp(x₂) = exp(x₁+x₂)

This follows from the uniqueness property of logarithms and the fundamental properties of logarithms.

The fundamental property of logarithms implies

• ln( 1/a) = (-1) ln(a) as
  0 = ln(1) = ln ( (1/a) a )
• ln(a ^m) = m ln (a) for all whole numbers and then for all integers. integers.

Logarithms to base c > 0.

The logarithm of x > 0 to a base c > 0 is given by

Here ln(e) = 1 implies log_e(x) = ln(x).

The logarithm of x > 0 to a base 10 is given by

The button log(x) on a calculator computes log₁₀(x).

The definition of log_c(x) in terms of ln(x) implies

• log_c(ab) = log_c(b) + log_c(a) for a> 0, b > 0 and c >0
• log_c( 1/a) = (-1) log_c(a) as  0 = log_c(1) = log_c( (1/a) a )
• log_c(a ^m) = m log_c (a)