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)

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