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:\({{\log }_{a}}x=\frac{\lg x}{\lg a}=\frac{\ln x}{\ln a.}\)