Natural Logarithms and Exponentials

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



Fundamental property of logarithms: ln(ab) = ln(b) +ln(a)

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.



a^m = exp( ln(a^m)) = exp(m ln (a))

a^1/m = exp( (1/m) ln(a) )

a^m/n = (a^m)^1/n = exp( (m/n) ln(a))