Logaritmi

    Definitie:Fie a>0,$a\ne 1$ si x>0.Unicul numar real y cu proprietatea ${{a}^{y}}=x$ se numeste logaritmul numarului x in baza a si se noteaza ${{\log }_{a}}x$.

    Cu alte cuvinte, ${{\log }_{a}}x$=y daca si numai daca ${{a}^{y}}=x$.

Observatii:1.Daca$a=10$,numarul ${{\log }_{10}}x=\lg x$ se numeste logaritmul zecimal al lui x.

                   2.Daca $a=e$,numarul ${{\log }_{e}}x=\ln x\,$se numeste logaritmul natural al lui x.

Proprietatile logaritmilor:

1. ${{a}^{{{\log }_{a}}x}}=x,\forall x>0$;

2.${{\log }_{a}}{{a}^{x}}=x,\forall x\in \mathbb{R}$;

3.${{\log }_{a}}a=1,\forall a>0,a\ne 1;$

4.${{\log }_{a}}1=0,\forall a>0,a\ne 1;$

5.${{x}^{{{\log }_{a}}y}}={{y}^{{{\log }_{a}}x}}$.

Operatii cu logaritmi:

1. ${{\log }_{a}}x+{{\log }_{a}}y={{\log }_{a}}\left( xy \right),\forall x,y>0$

2.${{\log }_{a}}x-{{\log }_{a}}y={{\log }_{a}}\left( \frac{x}{y} \right),\forall x,y>0$

3.${{\log }_{a}}{{x}^{p}}=p{{\log }_{a}}x,\forall x>0,\forall p\in \mathbb{R}$

4.${{\log }_{{{a}^{p}}}}x=\frac{1}{p}{{\log }_{a}}x,\forall x>0,\forall p\in {{\mathbb{R}}^{*}}$

Schimbarea bazei unui logaritm:

1.${{\log }_{a}}x=\frac{{{\log }_{b}}x}{{{\log }_{b}}a},\forall a,b,x>0;a,b\ne 1$

2.${{\log }_{a}}b\cdot {{\log }_{b}}c={{\log }_{a}}c,\forall a,b,c>0;a,b\ne 1$

3.${{\log }_{a}}b=\frac{1}{{{\log }_{b}}a};\forall a,b>0;a,b\ne 1$

Consecinta:${{\log }_{a}}x=\frac{\lg x}{\lg a}=\frac{\ln x}{\ln a.}$