exponentiation

A naive definition of exponentiation would be:

a^b:=∏\limits_{1}^{b} a

But what happens when b is not a natural number?

Then, a better definition is:

a^b:=\exp(b⋅\ln a)

where "exp" is:

\exp x:=\lim \limits_{n\rightarrow ∞} (1+÷{x}{n})^n
e:=\exp 1=\sum\limits_{n=0}^{∞} ÷{1}{n!}
\exp x=\sum\limits_{n=0}^{∞} ÷{x^n}{n!}

where "ln" (the "natural logarithm") is:

\exp(a)=b\Leftrightarrow \ln(b)=a

Properties of "exp"

  • \exp(x+y)=\exp(x)⋅\exp(y)
  • ÷{1}{\exp(x)}=\exp(-x)
  • \exp(x)>0
  • \exp(z+2⋅π⋅i)=\exp(z)
  • ÷{d}{dx}(\exp x)=\exp x
  • exp(x, x positive) rises very very quickly.

Other Logarithms

a^b=c\Leftrightarrow \log_a c=b

where:

\log_a x:=÷{\ln x}{\ln a}

Properties of (all) Logarithms

  • \log_a(x⋅y)=\log_a(x)+\log_a(y) (for positive x,y)
  • \log_a(÷{x}{y})=\log_a(x)-\log_a(y) (for positive x,y)
  • \log_a(x^y)=y⋅\log_a(x)

Author: Danny (remove the ".nospam" to send)

Last modification on: Sat, 04 May 2024 .