Eksponentielle og potens funktioner

                    ...har du ikke en matematikbog omhandlende logaritmer?

Jo det har jeg. Problemet er at jeg ikke forstår den og min lærer er utrolig dårlig til at lære fra sig. 

    eksponentiel funktion

                             voksende
                                                      f(x) = b·e^k·x      k∈R₊

                                                      f(x+X2) = b·e^k·(x+X2) = b·e^k·x · e^kX2 = e^kX2·f(x)

                                                      f(x+X2) / f(x) = 2 = e^kX2

                                                                               ln(2) = k·X2

                                                                               X2 = ln(2) / k

                             aftagende
                                                      f(x) = b·e^-k·x      k∈R₊

                                                      f(x+X½) = b·e^-k·(x+X½) = b·e^-k·x · e^-kX½ = e^-kX½·f(x)

                                                      f(x+X½) / f(x) = (1/2) = e^-kX½

                                                                               ln(1/2) = -k·X½

                                                                               -ln(2) = -k·X½

                                                                               X½ = ln(2) / k

eller skrevet

    eksponentiel funktion

                             voksende
                                                      f(x) = b·a^x      a>1

                                                      f(x+X2) = b·a^x+X2 = b·a^x · a^X2 = a^X2·f(x)

                                                      f(x+X2) / f(x) = 2 = a^X2

                                                                               ln(2) = ln(a)·X2

                                                                               X2 = ln(2) / ln(a)

                             aftagende
                                                      f(x) = b·a^x      0<a<1

                                                      f(x+X½) = b·a^(x+X½) = b·e^-k·x · a^X½ = a^X½·f(x)

                                                      f(x+X½) / f(x) = (1/2) = a^X½

                                                                               ln(1/2) = ln(a)·X½

                                                                               X½ = ln(1/2) / ln(a)

Hej mathon.

Vil du prøve at forklare det med ord? 

                                        ln(x)      Dm(ln) = R₊    Vm(ln) = R
regneregler:

                                        ln(a·b) = ln(a) + ln(b)

                                        ln(a/b) = ln(a) / ln(b)                   

                                        ln(aⁿ) = n·ln(a)

                                        ln(ⁿ√(a)) = (1/n)·ln(a)

Tak :D

          fordoblingskonstant:
 
                     når et vilkårligt x øges med X2, fordobles funktionsværdien

          halveringskonstant:
 
                     når et vilkårligt x øges med X½, halveres funktionsværdien